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\rhead{\small Texas Tech University, Joshua Mayer, December 2014}

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Multivariate Extension to the Exponentiated Weibull Distribution with Analysis of Confidence Regions.\\[15pt]  %Edit
by\\[15pt]
Joshua Spencer Mayer\\[15 pt]
Bachelor of Science, Mathematics\\
Texas Tech University, 2010\\[15 pt]
Bachelor of Science, Economics\\
Texas Tech University, 2010\\[15pt]   %Edit
A Thesis\\[15pt]   % or Thesis
In\\[15pt]
Mathematics and Statistics\\[10pt]
Submitted to the Graduate Faculty\\
of Texas Tech University in\\
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Master of Science\\[15pt]  %Edit
Approved\\[15pt]
Dr. James Surles\\[15pt]
Chair of Committee\\[15pt] %Edit
Dr. Leif Ellingson\\[15 pt] %Edit
December, 2014      %Edit
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\chapter{Acknowledgments}
I wish to acknowledge the help and support of Dr. James Surles. Without his help, much of this work would have never been able to be accomplished. I would also like to thank the members of my committee: Dr. Leif Ellingson.\\
I would also like to thank my colleagues who have helped and encouraged me in many ways. 

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\chapter{Abstract}
In many situations, it is of interest to construct a parametric model for a collection of dependent random variables that are not necessarily Gaussian. Methods of constructing multivariate models using both a one-to-one transformation and copulas are given. With these constructions, methods of sampling and inference are discussed. These methods are applied to a particular multivariate extension of the exponentiated Weibull distribution, where there is discussion of the adequacy of these procedures using empirical measures.
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\chapter{Introduction}
Waloddi Weibull (1951) found that the two parameter distribution that was to bear his name had a wide range of applicability. The two parameter Weibull distribution is given as follows:\\
We say a random variable $X$ has the Weibull distribution, denoted $X\sim W(\lambda,\alpha)$, if it has a cumulative distribution function (cdf) given by
\begin{equation}
F(x|\lambda,\alpha)=1-\exp[-(\lambda x)^\alpha],\,x>0,\lambda>0,\alpha>0.
\label{wcdf}
\end{equation}
The probability density function (pdf) of the two parameter Weibull distribution is given by
\begin{equation}
f(x|\lambda,\alpha)=\alpha\lambda(\lambda x)^{\alpha-1}\exp[-(\lambda x)^\alpha],\, x>0,\lambda>0,\alpha>0.
\label{wpdf}
\end{equation}
 Weibull (1951) found that the Weibull distribution is a valid distribution to model the probability of a failure of chain of $n$ links. It has been shown that the Weibull distribution is an extreme value distribution; that is, it is a limiting distribution for the minimum of a large collection of independent and identically distributed (i.i.d.) random variables (under certain conditions). This leads to it having a large applicability in reliability and survival analysis, which is simply the study of the probability that a system of components will survive. The two parameter Weibull distribution is a popular choice for this sort of analysis because of its extreme value properties. \\
 One thing to note is the single variate two parameter Weibull distribution is limited in the fact it cannot capture the correlation between two sets of data that may be Weibull. It is thus of interest to consider multivariate extensions to the Weibull distribution to serve as a model in situations where this is appropriate. How these extensions are defined vary. Some examples may be seen in Roy and Mukherjee (1996), Yeh (2009), Lu and Bhattacharyya (1990), and Lee (1979).\\
 In general, many nonnormal multivariate distributions have been constructed using copulas. The following definition was given by Nelson (2006).
 \begin{definition}
 A function, $C:[0,1]^p\rightarrow[0,1]$ is a \textbf{p-dimensional copula} if:
 \begin{enumerate}[(i)]
 \item $C(u_1,...,u_{i-1},0,u_{i+1},...,u_p)=0$, for each $i=1,...,p$, and
 \item $C(1,...,1,u_i,1,...,1)=u_i$ for each $i=1,...,p$, and
 \item For each hyperrectangle $\bm{B}=\bigotimes_{i=1}^p[x_i,y_i]\in[0,1]^p$
 \begin{center}$V_C(\bm{B})=\Delta_{x_p}^{y_p}\Delta_{x_{p-1}}^{y_{p-1}}...\Delta_{x_1}^{y_1}C(\bm{t})\geq0$,\end{center}
  where $\Delta_{x_i}^{y_i}C(\bm{t})=C(t_1,...,y_i,...,t_p)-C(t_1,...,x_i,...,t_p)$ .
 \end{enumerate}
 \end{definition}
 In other words, $C:[0,1]^p\rightarrow[0,1]$ is p-dimensional copula if $C(\cdot)$ is a joint cdf of the p-dimensional vector of random variables (random vector) on the unit hypercube that has marginals that are uniformly distributed on $[0,1]$.  
 To see how copulas relate to the joint distribution of a p-dimensional random vector, Sklar's theorem must be considered. (Sklar, A. 1959)
 \begin{theorem}[\textbf{Sklar's Theorem}] Let $F(\cdot)$ be a joint p-dimensional cdf for the random vector $(X_1,...,X_p)^\top$ with marginal distribution functions $F_1(\cdot),...,F_p(\cdot)$, respectively. Then there exists a p-dimensional copula $C(\cdot)$ such that for each $\bm{x}\in\mathbb{R}^p$,
 \begin{equation} F(x_1,...,x_p)=C\left(F_1(x_1),...,F_p(x_p)\right). \label{copp} \end{equation}
 Moreover, if $F_1(\cdot),...,F_p(\cdot)$ are each continuous, the function $C(\cdot)$ is unique. Conversely, if $C(\cdot)$ is a p-dimensional copula and $F_1(\cdot),...,F_p(\cdot)$ are the cdfs of each of the components of the random vector $(X_1,...,X_p)^\top$, then the function $F(\cdot)$ defined in \eqref{copp} is a p-dimensional cdf of the random vector $(X_1,...,X_p)^\top$ with marginal distribution functions $F_1(\cdot),...,F_p(\cdot)$, respectively. \label{sklar}
 \end{theorem}
 What this theorem states is that every joint distribution function, $F(x_1,...,x_p)$, of the random vector, $(X_1,...,X_p)^\top$, with marginal distribution functions $F_1,...,F_p$, is a copula with arguments $F_1,...,F_p$. More importantly, every  copula with arguments $F_1,...,F_p$ is a joint distribution function with marginal distribution functions $F_1,...,F_p$. Hence if we consider the random vector $(X_1,...,X_p)^\top$ such that if $X_i\sim W(\lambda_i,\alpha_i)$ with cdf $F_i(x_i)$ as in (\ref{wcdf}), $i=1,...,p$, and $C(\cdot)$ is a p-dimensional copula, then the function $C(F_1(x_1),...,F_p(x_p))$ is a joint distribution function with each marginal distribution being $W(\lambda_i,\alpha_i)$.\\
  One may note that in survival analysis the survival function is usually considered. This can be troubling, as it seems the use of copulas rely on using the cdf to construct a joint distribution, rather than the use of a survival function to construct a joint survival function. This can be alleviated by using the \emph{survival copulas}. A p-dimensional survival copula is a copula whose arguments are survival functions. Nelson (2006) notes that survival copulas ``couples the joint survival function to its univariate margins in a manner completely analogous to the way in which a copula connects the joint distribution function to its margins.'' Namely, there is an analog of Sklar's theorem for survival functions.\\
 Using this, a multivariate cdf can easily be constructed using copulas. Consider the random vector $\bm{X}=(X_1,...,X_p)^\top$ where the survival function of the $i^{th}$ component of that vector is given as $P(X_i> x_i)=\bar{F}_i(x_i)$. Then the joint survival function of $\bm{X}$ is defined to be, for each $\bm{x}\in\mathbb{R}^p$:
 \begin{equation} \bar{F}(\bm{x})=\hat{C}\left(\bar{F}_1(x_1),...,\bar{F}_p(x_p)\right),\label{cdfcop}\end{equation} where $\hat{C}:[0,1]^p\rightarrow[0,1]$ is a copula.\\
 Many multivariate two parameter Weibull joint distributions have been constructed using \eqref{cdfcop}, and hence a need to classify these multivariate distributions arose. Lee (1979) proposed a classification of the multivariate two parameter Weibull model which related multivariate distributions to the Weibull minimum property. Another form of classification was proposed by Roy (1994) for life distributions in multivariate models. These classifications are a generalization of the classification typically used for univariate lifetime models in survival and reliability analysis. Since this thesis is about more general multivariate life models, the Roy classification is more appealing. Before this classification is discussed, notation will be needed.\\
 \begin{definition} Let $\bm{X}=(X_1,...,X_p)^\top$ be a random vector with support, $Supp(\bm{X})\subset(0,\infty)^p$ with joint survival function $\bar{F}(\bm{x})$, and let $\bm{x}=(x_1,...,x_p)^\top\in\mathbb{R}^p$\\ \noindent
\textbf{The Cumulative Hazard Function} (CHF), $H(\bm{x})$, is defined to be:
  \begin{equation} H(\bm{x})=-\log(\bar{F}(\bm{x})).\label{chf}\end{equation}
  \textbf{The Hazard Function} (HF) for the $i^{th}$ coordinate of the random vector, $h_i(\bm{x})$, is defined to be:
  \begin{equation}h_i(\bm{x})=\frac{\partial H(\bm{x})}{\partial x_i}.\label{hf}\end{equation}
  \textbf{The Failure Rate Average} (FRA), of the $i^{th}$ coordinate, $\text{FRA}_i(\bm{x})$ is defined to be:
  \begin{equation} \text{FRA}_i(\bm{x})=\frac{1}{x_i}\int_0^{x_i}h_i(\bm{x}_{1:i:p}(u))\,du,\label{fra}\end{equation}
  where $\bm{x}_{1:i:p}(u)=(x_1,...,x_{i-1},u,x_{i+1},...,x_p)$.\\
  \textbf{The multivariate Mean Residual Life} (MRL) of the $i^{th}$ coordinate, $\text{MRL}_i(\bm{x})$ is defined to be:
  \begin{equation}\text{MRL}_i(\bm{x})=\mathbb{E}[X_i-x_i|\Omega_{1:i:p}(X_i>x_i)],\label{mrl}\end{equation} where $\Omega_{1:i:p}(\xi)=X_1>x_1,...,X_{i-1}>x_{i-1},\xi,X_{i+1}>x_{i+1},...,X_p>x_p$ where $\xi$ is a relation on $X_i$.
  \end{definition}
  With these definitions in mind, Roy (1994) used the following classification:\\
  \noindent Suppose the random vector $\bm{X}=(X_1,...,X_p)^\top$ has joint survival function $\bar{F}(\bm{x})$ and cdf $F(\bm{x})$.
  \begin{enumerate}[(i)]
  \item $\bm{X}$ is said to have a multivariate increasing (decreasing) failure rate, (MIFR(MDFR)) if $h_i(\bm{x})$ is increasing (decreasing) in $x_i$ for all $\bm{x}$ and for each $i=1,...,p$ 
  \item $\bm{X}$ is said to have a multivariate increasing (decreasing) failure rate average (MIFRA(MDFRA)) if $\text{FRA}_i(\bm{x})$ is increasing (decreasing) in $x_i$ for all $\bm{x}$ and for each $i=1,...,p$
  \item $\bm{X}$ is said to have the multivariate new better (worse) than used (MNBU(MNWU)) if $\frac{\bar{F}(\bm{x}_{1:i:p}(x_i+y_i))}{\bar{F}(\bm{x}_{1:i:p}(x_i))}\leq(\geq)\frac{\bar{F}(\bm{x}_{1:i:p}(y_i))}{\bar{F}(\bm{x}_{1:i:p}(0))}$ for all choices of $y_i\geq0$ and for all $\bm{x}$ and for each i.
  \item $\bm{X}$ is said to have the multivariate new better (worse) than used in expectation (MNBUE(MNWUE)) if
  $\text{MRL}_i(\bm{x})\leq(\geq)\mathbb{E}[X_i|\Omega_{1:i:p}(\emptyset)]$
  \item $\bm{X}$ is said to have multivariate increasing (decreasing) mean residual life (MIMRL(MDMRL)) if $\text{MRL}_i(\bm{x})$ is increasing (decreasing) in $x_i$ for all $\bm{x}$ and each $i$.
  \end{enumerate}
  Roy (1994) found the following relationship between these classes:
  \begin{enumerate}[(a)]\item MIFR$\subset$MIRFRA$\subset$MNBU$\subset$MNBUE\item MIFR$\subset$MDMRL$\subset$MNBUE\end{enumerate}
  Which is consistent with the univariate results found about this. \\
 One such multivariate distribution was given by Roy and Mukherjee (1996) for the multivariate two parameter Weibull distribution as the following:
\begin{equation}\bar{F}(\bm{x})=\exp\left[-\left\{\sum_{i=1}^p(\lambda_i^\nu x_i^{\alpha\nu})\right\}^{1/\nu}\right];\,\,x_i>0,\,\lambda_i>0\text{ for each } i,\,\alpha>0,\,\nu\geq 1.\label{rm}\end{equation}
 The parameter $\nu=1$ gives the joint distribution of independent Weibull random variables. Roy and Mukherjee (1996) have shown that this distribution is in the class of MIFR and MRIFRA. They have also shown that $Z=\underset{1\leq i \leq p}{\min}X_i$ maintains the Weibull minimum property, that is, $Z\sim W(\lambda,\alpha)$ where $\lambda=(\sum_{i=1}^p\lambda^\nu)^{1/\nu}$.\\ 
Another major weakness the two parameter Weibull distribution exhibits is that its hazard function can only accommodate monotone hazard functions. For a univariate random variable, \eqref{hf} can be interpreted as the instantaneous rate in which a component fails at a time period $t$.\\
 We can see that if $X\sim W(\lambda,\alpha)$, then $h(x)=\alpha\lambda^\alpha x^{\alpha-1}$, which is a strictly monotone increasing function of $x$ if $\alpha>1$, a strictly monotone decreasing function of $x$ if $\alpha<1$, and it is a constant function of $x$ if $\alpha=1$ (which is the exponential distribution). For example, as discussed by Pokharel (2011),  this could accommodate data for automotive vehicles. As an automotive vehicle grows older, its hazard function should increase, \emph{i.e.}  the rate at which automotive vehicles fail should increase as they grow older. This would lead us to believe that the two parameter Weibull may be an appropriate model for this process. However, considering the hazard function for the lifetime of human beings (given by Pokharel (2011)), we would expect people to die more frequently earlier in life, since infants are more susceptible to disease, and tend to die more frequently later in life due to the perils of old age. We can see that this would lead us to believe that this hazard function seems to be more of a ``bathtub'' shaped hazard function (a hazard function whose values are higher at lower values and higher values of $x$, and lower at values of $x$ that are in the middle) and would lead us to believe that the two parameter Weibull distribution may not be an appropriate model.\\
  This lead to the need for a generalization that can accommodate nonmonotone hazard functions. Mudholkar and Srivastava (1993) introduced a family of distributions that are able to accommodate nonmonotone and bathtub shaped hazard functions. They called this family of distributions the Exponentiated Weibull (EW) distribution. This distribution can be obtained by exponentiating the cdf of two parameter Weibull distribution.\\
It is said $X$ has an exponentiated Weibull distribution with parameters $\lambda$, $\alpha$, and $\theta$ (denoted $X\sim \text{EW}(\lambda,\alpha,\theta)$) if it has cdf
\begin{equation}
F(x|\lambda,\alpha,\theta)=\{1-\exp[-(\lambda x)^\alpha]\}^\theta,\,x>0,\lambda>0,\alpha>0,\theta>0.
\label{ewcdf}
\end{equation}
The probability density function (pdf) is given by:
\begin{equation}
f(x|\lambda,\alpha,\theta)=\alpha\theta\lambda(\lambda x)^{\alpha-1}\exp[-(\lambda x)^\alpha]\{1-\exp[-(\lambda x)^\alpha]\}^{\theta-1},\,x>0,\lambda>0,\alpha>0,\theta>0.
\label{ewpdf}
\end{equation}
Using \eqref{ewcdf} and \eqref{ewpdf}, Nadarajah \emph{et al.} (2013) found the hazard function to be:
\begin{equation}
h(x)=\frac{f(x)}{\bar{F}(x)}=\alpha\theta\lambda(\lambda x)^{\alpha-1}\exp[-(\lambda x)^\alpha]\{1-\exp[-(\lambda x)^\alpha]\}^{\theta-1}.
\label{ewh}
\end{equation}
Using \eqref{ewh}, it was shown by Mudholkar and Srivastava (1993) and Nadarajah \emph{et al.} (2013) that this distribution accommodates constant, monotonic, decreasing, increasing, bathtub and unimodal hazard functions. In particular for $\theta=\alpha=1$ \eqref{ewh} is a constant hazard function (this gives the hazard function for the exponential distribution). For $\theta=1$ and $\alpha\neq1$, it is monotone increasing or decreasing (this gives the hazard function for the two parameter Weibull). It gives a bathtub shape when $\alpha>1$ and $\theta\alpha<1$. It gives a unimodal shape when $\alpha<1$ and $\theta\alpha>1$. It also gives a monotone increasing (decreasing) shape when $\alpha>(<)1$ and $\theta\alpha>(<)1$. This wide range of hazard function shapes leads it to be a suitable model for many survival processes.\\
Nassar and Eissa (2003) have found that there is indeed a relationship between the hazard function and the mean residual life (MRL) of the exponentiated Weibull. Nassar and Eissa have shown that if $X$ is an EW distribution, \eqref{mrl} reduces to:\\
\begin{equation}\text{MRL}(t)=(\mathbb{E}[X]-I(t)-t)/\bar{F}(t),\label{ewmrl}\end{equation}
where $I(t)=\int_0^tF(x)\,dx$. Nassar and Eissa have shown that \eqref{ewmrl} is constant when \eqref{ewh} is constant, decreasing (increasing) when \eqref{ewmrl} is increasing (decreasing) and upside down bathtub shaped (bathtub shaped) when \eqref{ewh} is bathtub shaped (upsidedown bathtub shaped). Nassar and Eissa have shown it follows that, using similar notation to the Roy categorization, $\text{MRL}(t)$ is DMRL (IMRL) if $\alpha\geq(\leq)1$ and $\alpha\theta\geq(\leq)1$. $\text{MRL}(t)$ is bathtub shaped (BMRL) (upsidedown bathtub shaped (UMLR)) if $\alpha<(>)1$ and $\alpha\theta>(<)1$.\\
There have been extensive studies of the exponentiated Weibull distribution. The following is a review of the literature of the exponentiated Weibull distribution. A more extensive review can be found in Nadarajah \emph{et al.} (2013). Many of the various properties of the EW distribution have been found in Mudholkar and Srivastava (1993), Mudholkar \emph{et al}. (1995), and Mudholkar and Hutson (1996). These properties are outlined in Nadarajah \emph{et al.} (2013).\\
The EW distribution is also a generalization of many other distributions. Of course, for $\theta=1$ this reduces to the two parameter Weibull distribution. For $\alpha=1$, this generalizes as the Exponentiated Exponential distribution, given by Gupta and Kundu (1999). For $\alpha=2$ and $\lambda=1$, this becomes the Burr type X distribution, developed by Burr (1942). For $\alpha=2$, this becomes the scaled Burr type X, which was studied extensively by Surles and Padgett (2001) and Surles (1999) among others (see Nadarajah \emph{et al.} (2013)).\\
Mudholkar \emph{et al.} (1995) derived the kth order moment for the univariate exponentiated Weibull when $\theta$ is an integer, as \[ \mathbb{E}[X^k]= \lambda^{-k}\theta\left(\frac{k}{\alpha}+1\right)\sum_{j=0}^{\theta-1}(-1)^j\binom{\theta-1}{j}(j+1)^{-k/\alpha-1}. \]
Choudhury (2005) and Nadarajah and Gupta (2005) independently derived of the kth order moments $(k>-\alpha)$ when $\theta$ is a positive real number as
\begin{equation}
\mathbb{E}[X^k]=\theta\lambda^{-k}\Gamma\left(\frac{k}{\alpha}+1\right)\sum_{i=0}^\infty\frac{(1-\theta)_i}{i!(i+1)^{(n+\alpha)/\alpha}};\label{moments}
\end{equation}
where \[
(1-\theta)_i=\prod_{j=0}^{i-1}(\theta+j).
\]
Nadarajah \emph{et al.} (2013) note that the infinite series converges for all $\theta>0$. 
In particular, if $\theta\geq1$ is an integer, Nassar and Eissa (2003) have found a simplified version of the $k$-th moment to be
\[
\mathbb{E}[X^k]=\theta\lambda^{-k}\Gamma\left(\frac{k}{\alpha}+1\right)\sum_{i=0}^{\theta-1|}\frac{(1-\theta)_i}{i!(i+1)^{(k+\alpha)/\alpha}}
\]
for each $k>-\alpha$.\\
\section{Some Further Works on the Exponentiated Weibull}
Salem and Abo-Kasen (2011) found Bayes estimators under hybrid censoring using scale-invariant priors for $\alpha$ and $\theta$ under squared error loss, and found the equations to obtain the maximum likelihood estimators (MLEs) under hybrid censoring and found the form of the elements of the Fisher information matrix.\\
Nassar and Elissa (2004) found the equations to find the MLEs under type II censoring for the two parameter exponentiated Weibull (and exponentiated Weibull with $\lambda=1$) when $\theta$ is known and when $\theta$ is unknown. They also found the Bayesian estimators for a two parameter exponentiated Weibull with $\theta$ known using the gamma conjugate prior for $\alpha$ with respect to squared error loss and LINEX loss, and found the Bayesian estimators of the survival function and the hazard function and Bayseian estimators for $\alpha$ and $\sigma$ unknown using prior conditional on $\alpha$ to be of a Gamma form, and the prior of $\alpha$ to be exponential. Using this, they found Bayseian estimators for $\alpha$ and $\sigma$.\\
Cancho and Bolfarine (2001) create a exponentiated Weibull mixture model and then perform inference using maximum likelihood and then find the equations to obtain maximum likelihood estimators and perform model adequacy checks using likelihood ratio statistics. They also do Bayseian inference on the model, using gamma and Jefferys priors, by implementing MCMC methods and discuss methods of model selection.\\
Nassar and Eissa (2003) find many properties of the exponentiated Weibull, including simplified versions of the moments, quartile functions and modes. They also found properties of the hazard function and the mean residual life function.\\
 Nadarajah and Kotz (2006) find other simplifications of moments of the exponentiated Weibull. Ali \emph{et al.} (2006) defined some other exponentiated distributions. Jiang and Murthy (1999) found a graphical approach to the exponentiated Weibull and found other methods of estimating the parameters. Qian (2011) found methods of finding maximum likelihood estimators. Mudholkar and Srivastava (1993) found methods of obtaining maximum likelihood estimators of the exponentiated Weibull along with goodness of fit tests. Cordeiro \emph{et al.} (2011) found other closed form moments and also found the components of the information matrix of the exponentiated Weibull. Ashour and Afify (2007) found methods of finding the MLEs and the information matrix under type I progressive interval censoring. Cheng \emph{et al.} (2010) propose an algorithm for obtaining the maximum likelihood estimators for progressive type I censored data. Singh \emph{et al.} (2002) have found methods of estimating the exponentiated Weibull shape parameters under Linex loss, then compare their risks under LINEX loss to the MLEs, generalized MLEs and Bayes estimators under squared error loss. Mudholkar and Hutson (1996) derived classifications of the exponentiated Weibull's hazard function and density function based on the parameters.\\
 \section{Thesis Outline and Purpose}
The purpose of this thesis is to develop methods of constructing multivariate extensions of the exponentiated Weibull distribution. Many have been proposed (Sabina and Castillo (2005), Surles and Padgett (2005), Kundu and Gupta (2010), and Kundu and Gupta (2011)), and as such, it is of interest to develop, methods of constructing these distributions, and when considering a specific distribution, methods for constructing asymptotic tests and confidence regions. The means of how this will be discussed is as follows: \\
In Chapter II, methods of constructing such distributions will be discussed, and then a particular model will be considered. Chapter III will discuss methods of sampling from these distributions and computing estimators for the parameters. It will then discuss methods of constructing confidence regions using asymptotic results and bootstrapping methods. Chapter IV will present a simulation study in which the confidence levels are observed and comparisons of the methods are considered. Chapter V will discuss conclusions of this thesis and summarize the results obtained.


\chapter{An Extension to the Multivariate Weibull Distribution}
\section{Introduction}
This chapter will discuss methods of constructing a multivariate extension to the exponentiated Weibull distribution, and then examine some simple properties of these distributions. It will then consider a particular distribution, in which further properties will be discussed.
\section{Methods of construction Multivariate extensions to the exponentiated Weibull distribution}
\subsection{A transformation approach}  
Throughout the literature, there have been many well developed multivariate two-parameter Weibull distributions. From this fact, such distributions can be used to construct a multivariate exponentiated Weibull distribution. To see how this construction works in general, the following proposition should be considered.
\begin{proposition} Suppose $X\sim W(\beta,k)$, and let 
\begin{equation}
Y=\frac{1}{\lambda}\left(-\log\left(1-\left(\exp\left(\left(-X\beta\right)^k\right)\right)^{1/\theta}\right)\right)^{1/\alpha}, \lambda>0,\alpha>0,\theta>0.
\label{trans}
\end{equation} Then $Y\sim EW(\lambda,\alpha,\theta)$\label{transform}.\end{proposition}
\begin{proof} $F(y)=P(Y\leq y)=P\left(X>-\frac{1}{\beta}\left(\log\left(\left(1-\exp\left(-(\lambda y)^\alpha\right)\right)^\theta\right)\right)^{1/k}\right)=(1-\exp(-(\lambda y)^\alpha))^\theta$. \end{proof}

From this, it is clear that a class of multivariate exponentiated Weibull distribution can be created from this very basic idea. It should be first noted that it is customary to define  a joint distribution, $F(x_1,x_2,...,x_p)$ to be a multivariate extension to a distribution, $F_0(x|\bm{\theta})$, $\bm{\theta}\in\Theta$, where $\Theta\subset\mathbb{R}^{d}$, if each of the marginal distributions of the joint distribution are distributed as $F_0(x|\bm{\theta}_i)$,  $\bm{\theta}_i\in\Theta$, \emph{i.e.} if $(X_1,...,X_p)^\top\sim F(\bm{x})$ and $X_i\sim F_0(x|\bm{\theta}_i),~\bm{\theta}_i\in\Theta$ for $i=1,...,p$, then $F(\cdot)$ is a multivariate extension to $F_0(\cdot|\bm{\theta})$. By using this, a multivariate extension to the exponentiated Weibull distribution can be constructed from a multivariate extension of the Weibull distribution as follows.
\begin{theorem}
Let $\bm{X}=(X_1,...,X_{p})^\top$ have joint distribution $F(\bm{x})$, such that for each $i\in(1,...,p)$, $X_i$ has marginal distribution function of the form of (\ref{wcdf}). For each $i\in(1,...,p)$, let $Y_i=\frac{1}{\lambda_i}\left(-\log\left(1-\left(\exp\left(\left(-X_i\beta_i\right)^{k_i}\right)\right)^{1/\theta_i}\right)\right)^{1/\alpha_i}\, \lambda_i>0,\alpha_i>0,\theta_i>0$. Then $(Y_1,...,Y_p)^\top$ has the property that for each $i\in (1,...,p)$, $Y_i$ has pdf of the form (\ref{ewpdf}).  
\end{theorem}
\begin{proof} Immediate from Proposition \ref{trans}.
\end{proof}
\\
From here, it can be seen that constructing a joint distribution for an exponentiated Weibull can be reinterpreted as constructing a joint distribution for a Weibull and then transforming the components using Theorem \ref{trans}.
\subsection{A Copula Approach}
Another approach that may be taken to construct a multivariate extension to the exponentiated Weibull distribution is to use a copula (constructed or preexisting) to develop a multivariate exponentiated Weibull distribution. It is known from Sklar's theorem (Theorem \ref{sklar}) that if $\bm{X}=(X_1,...,X_p)^\top$ in which $X_i\sim EW(\lambda_i,\alpha_i,\theta_i)$ for $i=1,...,p$ with cdf $F_i(x_i)$ (as defined in (\ref{ewcdf})), then there is a copula $C(\bm{u})$, $\bm{u}\in[0,1]^p$, such that for each $\bm{x}\in(0,\infty)^p$, $P(\bm{X}\leq\bm{x})=C(F_1(x_1),...,F_p(x_p))$. Thus, if $\bm{X}$ is defined as above, then it is of interest to define a copula, $C(\bm{u}|\bm{\nu})$, $\bm{u}\in[0,1]^p$ and $\bm{\nu}$ a collection of dependence parameters. Then, the joint distribution of $\bm{X}$ can be defined, for each $\bm{x}\in(0,\infty)^p$, as $F(\bm{x})=P(\bm{X}\leq\bm{x})=C(F_1(x_1),...,F_p(x_p)|\bm{\nu})$, in which the distribution defined by $F(\cdot)$ has the property that each $X_i\sim EW(\lambda_i,\alpha_i,\theta_i)$ for $i=1,...,p$. Hence we may call such a distribution a multivariate extension to the exponentiated Weibull distribution. \\
It can be seen that both methods show that there is a large variety of models that can be called multivariate extensions to the exponentiated Weibull distribution. 
\section{A particular Multivariate exponentiated Weibull distribution}
One multivariate extension that will be considered will be the one developed by Roy and Mukherjee (1998). Let $\bm{X}=(X_1,...,X_p)^\top$ be a random vector of dimension $p$, with each component having marginal distribution function $F_i(x_i)$, $i=1,...,p$. As before, let $\bar{F}_i(x_i)=1-F_i(x_i)$ be the survival function of the $i^{th}$ component and let $R_i(x_i)=-\log\bar{F}_i(x_i)$ be the hazard function of the $i^{th}$ component, $i=1,...,p$. Then the joint survival function, $\bar{F}(x_1,...,x_p)$, can be defined to be
\begin{equation}
\bar{F}(x_1,...,x_p)=\exp\left[-\left\{\sum_{i=1}^p(R_i(x_i))^\nu\right\}^{1/\nu}\right],
\label{surv}
\end{equation}
where $\nu\geq1$ is a dependence parameter.\\
It has been shown by Roy and Mukherjee (1998) that this is a valid multivariate extension, in which the marginal survival functions will be $\bar{F}_i(x_i)$. Further, they have derived the pdf of $\bm{X}=(X_1,...,X_p)^\top$ to be 
\begin{align}
f(x_1,...,x_p|\nu)=\left\{\prod_{i=1}^pR_i^{\nu-1}(x_i)r_i(x_i)\right\}\exp\left[-\left\{\sum_{i=1}^p(R_i(x_i))^\nu\right\}^{1/\nu}\right] \\
\times\left\{\sum_{i=1}^p a_{ip}\left(\sum_{i=1}^pR_i^\nu(x_i)\right)^{i(1-\nu)/\nu-(p-i)}\right\}, \nonumber
\end{align}
\label{genpdf}
where $r_i(x)=\frac{d}{dx}R_i(x)$ and $a_{ip}$ is defined by the equations
\begin{align*}
&a_{ik}=a_{(i-1)(k-1)}+a_{i(k-1)}((k-1)\nu-i),\,\,\, & i=2,3,...,k-1\\
&a_{kk}=1,\, a_{1k}=(\nu-1)...((k-1)\nu-1), & k=2,...,p.
\end{align*}
Although one could easily just let $X_1,...,X_p\sim EW(\lambda_i,\alpha_i,\theta_i)$, $i=1,...,p$ and then compute $R_i(x_i)=-\log(1-(1-\exp(-(\lambda_ix_i)^{\alpha_i}))^{\theta_i})$. This would of course cause a messy distribution, and hence it would be of interest to not look at the joint survival function but instead the joint distribution function. Thus the two methods of construction should be considered.
\subsection{Method 1: The Transformation Approach}
It has been shown by Roy and Mukherjee (1998) that if $Y_1,...,Y_p\sim W(\lambda_i,\alpha)$, $i=1,...,p$ then the joint survival function can be written as
\[ \bar{F}(y_1,....,y_p)=\exp\left[-\left\{\sum_{i=1}^p(\lambda_iy_i^{\alpha\nu})\right\}^{1/\nu}\right],
\]
where $\lambda_i>0$ for each $i\in\{1,...,p\}$, $\alpha>0$, and $\nu\geq1$.\\ 
As such, we can see that this yields a multivariate Weibull survival function. Hence, using Theorem 2.1, we may construct a multivariate Exponentiated Weibull distribution the following way. \\
Let $Y_1,...,Y_p\sim F(x)=1-\exp(-y)I_y(0,\infty)$, the exponential distribution with unit mean. Let the random vector $(Y_1,...,Y_n)^\top$ have a joint survival function of the form (\ref{surv}). Then $Y_i\sim W(1,1)$ and hence let 
\[
X_i=\frac{1}{\lambda_i}\left(-\log\left(1-\left(\exp\left(\left(-Y_i\right)\right)\right)^{1/\theta_i}\right)\right)^{1/\alpha_i},
\]
where $\lambda_i>0$, $\alpha_i>0$, and $\theta_i>0$ for each $i\in\{1,...,p\}$.\\
Then $(X_1,...,X_p)^\top$ has a distribution function $F(x_1,...,x_p)$ such that each $X_i\sim EW(\lambda_i,\alpha_i,\theta_i)$ for $i\in\{1,...,p\}$. 
\subsection{Method 2: The Copula Approach}
It can be seen that the joint survival function in (\ref{surv}) can be represented as
\[
\bar{F}(x_1,x_2,...,x_p)=\exp\left[-\left\{\sum_{i=1}^p(-\log\bar{F}(x_i))^\nu\right\}^{1/\nu}\right],
\]
in which it may be inferred that the copula corresponding to this joint survival function is
\begin{equation}
C(u_1,...,u_n)=\exp\left[-\left\{\sum_{i=1}^p(-\log u_i)^\nu\right\}^{1/\nu}\right].
\label{cop}
\end{equation}
This is a valid Copula (see Nelson (2006)). Hence, a multivariate extension to the exponentiated Weibull can be written as 
\begin{equation}
F(x_1,x_2,...,x_p)=\exp\left[-\left\{\sum_{i=1}^p(-\log F_i(x_i))^\nu\right\}^{1/\nu}\right],
\label{copeval}
\end{equation}
where $F_i(x_i)$ is given in  (\ref{ewcdf}) with parameters $\lambda_i$, $\alpha_i$, and $\theta_i$, $i=1,...,p$. Using this method, it can be ensured that if $(X_1,...,X_p)$ has joint cdf as in (\ref{copeval}), then $X_i$ has marginal distribution function $F_i(x_i)$. \\
In both cases, it can be seen that a multivariate extension to the exponentiated Weibull distribution has been found. From here, it is of interest to look at properties of this particular multivariate exponentiated Weibull distribution.
\section{Properties of the particular multivariate Weibull distribution}
\begin{definition} We say $\bm{X}=(X_1,...,X_p)^\top$ has a \textbf{multivariate exponentiated Weibull distribution} with parameters $\bm{\lambda}\in(0,\infty)^p$, $\bm{\alpha}\in(0,\infty)^p$, $\bm{\theta}\in(0,\infty)^p$, and $\nu\in[1,\infty)$ (denoted $\bm{X}\sim MVEW(\bm{\lambda},\bm{\alpha},\bm{\theta},\nu)$) if the cdf of $\bm{X}$ is of the form
\begin{equation}
F(x_1,...,x_p|\bm{\lambda},\bm{\alpha},\bm{\theta},\nu)=
\exp\left[-\left\{\sum_{i=1}^p(-\log F_i(x_i|\lambda_i,\alpha_i,\theta_i))^\nu\right\}^{1/\nu}\right],
\label{mvewcdf}
\end{equation}
where $F(x_i|\lambda_i,\alpha_i,\theta_i)$ is given in (\ref{ewcdf}).
\end{definition}  
From here, some basic properties of this distribution can be established.
\subsection{Dependence Structure for the Multivariate Exponentiated Weibull Distribution}
\begin{definition} Let $C(u_1,u_2,...,u_p)$ be a copula. $C(\cdot)$ is said to have \textbf{positive quadrant dependence} (pqd) if for each $p\geq 2$
\[
C(u_1,...,u_p)\geq \prod_{i=1}^pu_i.
\]
A multivariate distribution $F(x_1,...,x_p)$ (or equivalently $\bar{F}(x_1,...,x_p)$) is said to have positive quadrant dependence if its associated copula has positive quadrant dependence.
\end{definition}
\noindent Clearly  $F(\cdot)$ has positive quadrant dependence if and only if  $F(x_1,...,x_p)\geq\prod_{i=1}^pF_i(x_i)$ and $\bar{F}(x_1,...,x_p)\geq \prod_{i=1}^p\bar{F}_i(x_i)$.\\
 Positive quadrant dependence is a special type of dependence in which if it is known $F(\cdot)$ has pqd, the it can be inferred $P(X_i>x_i|X_j>x_j)\geq P(X_i>x_i)$ for each $i,j=1,...,p$. 
\begin{theorem} 
$F(x_1,...,x_p)$ as defined in (\ref{mvewcdf}) has positive quadrant dependence.
\label{pqd}
\end{theorem}  
\begin{proof} Applying Jensen's inequality, the function $\rho_\nu(x)=x^{1/v}$ is concave for $\nu\geq1$, hence
\begin{align*}
F(x_1,...x_p)&=\exp\left[-\left\{\sum_{i=1}^p(-\log F(x_i))^\nu\right\}^{1/\nu}\right] \\
 & \geq \exp\left[\sum_{i=1}^p\log(F_i(x_i))\right]\\
 & = \prod_{i=1}^pF_i(x_i).
\end{align*} 
\end{proof}
\begin{theorem}
Let $(X_1,...,X_p)^\top$ be a random vector with joint distribution $F(x_1,...,x_p)$, and let $X_i$ have marginal distribution function $F_i(x_i)$, for $i=1,...,p$. Further, let $F(x_1,...,x_p)$ have positive quadrant dependence and let $X_i$ be such that $F_i(0)=0$ for $i=1,...,p$. Then $\text{Cov}(X_i,X_j)\geq 0$.
\end{theorem}
\begin{proof}
\begin{align*} 
\mathbb{E}[X_iX_j]&=\int_0^\infty\int_0^\infty x_ix_jf(x_i,x_j)\,dx_idx_j\\
&=\int_0^\infty\int_0^\infty\int_0^{x_i}\int_0^{x_j}f(x_i,x_j)\,dt_idt_jdx_idx_j\\
&=\int_0^\infty\int_0^\infty\int_{t_i}^\infty\int_{t_j}^\infty f(x_i,x_j)\,dx_idx_jdt_idt_j\\
&=\int_{0}^\infty\int_{0}^\infty\bar{F}(t_i,t_j)\,dt_idt_j\\
&\geq \int_0^\infty\int_0^\infty\bar{F}(t_i)\bar{F}(t_j)\,dt_idt_j\\
&= \mathbb{E}[X_i]\mathbb{E}[X_j]
\end{align*}
\end{proof}
The following result was proved in Roy and Mukherjee (1998).
\begin{corollary}
The distribution function defined in (\ref{mvewcdf}) is such that $\text{Cov}(X_i,X_j)\geq 0$ for each $i,j=1,...,p$. 
\end{corollary}
\section{Applications to Survival Analysis}
Previously, this paper has discussed methods for classifying multivariate distributions with respect to basic results in survival analysis. Roy and Mukherjee (1998) have been able to derive which classifications the distribution discussed in (\ref{surv}) fall into. These classifications will be discussed (briefly), then applied to the particular case of the exponentiated Weibull distribution. The fallowing two theorems were derived by Roy and Mukherjee (1998).
\begin{theorem}
Let $(X_1,...,X_p)\sim F(\bm{x})$ as defined in (\ref{surv}), and let $X_i\sim F_i(x_i)$. If $F_i(x_i)$ has IFR, then $F(\bm{x})$ has $MIFR$.
\end{theorem}
\begin{theorem}
Let $(X_1,...,X_p)\sim F(\bm{x})$ as defined in (\ref{surv}), and let $X_i\sim F_i(x_i)$. If $F_i(x_i)$ has IFRA, then $F(\bm{x})$ has $MIFRA$.
\end{theorem}
From these two theorems, the following results can be derived easily.
\begin{corollary} Let $\bm{X}=(X_1,...,X_p)^\top \sim MVEW(\bm{\lambda},\bm{\alpha},\bm{\theta},\nu)$. Then if $\alpha_i>1$ and $\theta_i\alpha_i>1$, then the distribution of $\bm{X}$ has MVIFR.
\end{corollary}
\section{Discussion of the Parameters}
In this section, the effect the parameters have on the shape of the underlying pdf is examined. To do this, different values of the parameters will be chosen and a graphical representation of the pdf for those parameters will be given, in terms of the univariate graph and corresponding contour plots for different values of the dependence parameter $\nu$. The first set of graphs that will be looked at will be the relationship of the two shape parameters

\begin{figure}[H]
\includegraphics[scale=0.8]{bathtubplot1-15-5}
\caption{PDF with $\lambda=1$, $\alpha=1.5$, $\theta=0.5$ (bathtub shaped hazard function)}
\label{bathtub}
\end{figure}
\begin{figure}[H]
\includegraphics[scale=0.8]{unimodal-1-5-3}
\caption{PDF with $\lambda=1$, $\alpha=.5$, $\theta=3$ (unimodal hazard function)}
\label{unimodal}
\end{figure}


\begin{figure}[H]
\includegraphics[scale=0.8]{monotonein-1-2-2}
\caption{PDF with $\lambda=1$, $\alpha=2$, $\theta=2$ (monotone increasing hazard function)}
\label{monotonein}
\end{figure}

\begin{figure}[H]
\includegraphics[scale=0.8]{monotonede-1-5-5}
\caption{PDF with $\lambda=1$, $\alpha=0.5$, $\theta=0.5$ (monotone decreasing hazard function)}
\label{monotonede}
\end{figure}


In these situations, it can be seen that small values of the parameter $\theta$ coincide with a monotone decreasing pdf (Figure \ref{bathtub}, Figure \ref{monotonein}), whereas larger values of $\theta$ coincide with a unimodal pdf (Figure \ref{unimodal}, Figure \ref{monotonein}). The values of $\alpha$ determine the location point of inflection of the pdf. Larger values of alpha will tend to move the point of inflection away from the origin whereas smaller values will move the point of inflection away from the origin. Larger values of the parameter $\nu$, as noted before, add a larger linear dependence, and the pdf will have more mass where the coordinates $x_1$ and $x_2$ are closer together. Since, as most lifetime distributions seem to behave, moves most of the mass of the pdf toward the origin.\\
The parameter $\lambda$ is a scale parameter, where larger values of $\lambda$ will cause a larger slope, and cluster more of the mass of the pdf toward the origin.
\chapter{Simulation, Estimation, and Inference}

\section{Methods of Obtaining a Sample from the Multivariate Extension to the Exponentiated Weibull}
In order to evaluate methods based on a distribution, a simulation study is typically considered. In this section, methods of obtaining a sample will be discussed. Two separate methods will be considered, and a discussion of these methods will be given. 
\subsection{The Gibbs Sampler}
In most multivariate distributions, a Gibbs sampler, named after Josiah Willard Gibbs, is typically the most standard and practical method of sampling from a multivariate distribution. If $(X,Y)$ have joint cdf $F(x,y)$ with marginal cdf of $Y$, $F(y)$, and the cdf of $X$ conditioned on $Y=y$ is $F(x|y)$, then it is known that $F(x,y)=F(x|y)\cdot F(y)$. From this, the Gibbs sampler was developed to provide an approximate sample to the multivariate distribution $F(x,y)$. \\
Suppose $(X_1,X_2,...,X_p)^\top$ have joint cdf  $F(x_1,x_2,...,x_P)$ and $X_i$ has marginal cdf $F_i(x_i)$ for $i=1,...,p$. A sample of size $n$ from the distribution $F(x_1,...,x_p)$ will be denoted $\bm{x}_{1},...,\bm{x}_n$, where the $j^{\text{th}}$ observation will be denoted $\bm{x}_j=(x_{1j},...,x_{pj})^\top$, $j=1,....,n$. The Gibbs sampler algorithm is as follows
 
\begin{algorithm}[H]
\KwResult{Generating an approximate sample of size $n$ from the joint distribution $F(x_1,...,x_p)$}
\For{$j=1$ to $n$}{
\For{$i=1$ to $p$}{
Simulate $x_{ij}^{(0)}$ from $F_i(x_i)$\;
} 
\For{$k=1$ to $m$}{
\For{$i=1$ to $p$}{
Simulate $x_{ij}^{(k)}$ from $F(x_i|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_{pj}^{(k-1)})$\;
}
}
Set $\bm{x}_j=(x_{1j}^{(m)},x_{2j}^{(m)},...,x_{pj}^{(m)})$\;
}
\caption{Gibbs Sampler for multivariate distribution}

\end{algorithm}
\[
\]
In order to perform the Gibbs sampler directly, it is first needed to sample from $F(x_i|x_{1},...,x_{i-1},x_{i+1},...,x_p)$. This can be problematic as the conditional cdf is usually is hard to obtain directly via integration. What must be done instead is to use copula theory, where the Gibbs sampler will be performed on the underlying copula and then a probability integral transformation will be used to transform the variables to ensure they come from the parent distribution. The following results will illustrate this fact.\\
This result is discussed by Remillard (2013).
\begin{proposition}
Let $(X_1,X_2,...,X_p)^\top $ be a continuous random vector with absolutely continuous joint distribution $F(x_1,...,x_p)$, and marginal distributions $F_i(x_i)$, $i=1,...,p$, in which $F(\cdot)$ has underlying copula $C(u_1,...,u_p)$ such that $F(x_1,....,x_p)=C(F_1(x_1),....,F_p(x_p))$. Then 
\begin{enumerate}[(i)]
\item $C(u_i|u_1,u_2,...,u_{i-1},u_{i+1},...,u_p)=\frac{\partial_{u_1}...\partial_{u_{i-1}}\partial_{u_{i+1}}...\partial_{u_p}C(u_1,...,u_p)}{\partial_{u_1}...\partial_{u_{i-1}}\partial_{u_{i+1}}...\partial_{u_p}C(u_1,...u_{i-1},1,u_{i+1},...,u_p)}$, $i=1,....,p$, where $C(u_i|u_1,u_2,...,u_{i-1},u_{i+1},...,u_p)=P(U_i\leq u_i|U_1=u_1,....,U_{i-1}=u_{i-1},U_{i+1}=u_{i+1},....,U_p=u_p)$
\item $F(x_i|x_1,...,x_{i-1},x_{i+1},....,x_p)=C(F(x_i)|F_1(x_1),....,F_{i-1}(x_{i-1}),F_{i+1}(x_{i+1}),...,F_p({x_p}))$ for $i=1,...,p$.
\end{enumerate}
\end{proposition}
In the bivariate case ($p=2$), this reduces to $C(u_i|u_j)=\partial_{u_j}C(u_1,u_2)$, and $F(x_i|x_j)=C(F_i(x_i)|F_j(x_j))$ where $i\neq j$, $i,j=1,2$. From this, a Gibbs sampler based on the underlying copula can be applied.\\
Again, let $\bm{X}=(X_1,...,X_p)^\top$ be a continuous random vector with absolutely continuous joint cdf $F(x_1,....,x_p)$ and $X_i$ have marginal cdf $F_i(x_i)$ for $i=1,...,p$ and let $F(\cdot)$ have underlying copula $C(u_1,...,u_p)$ satisfying $F(x_1,...,x_p)=C(F_1(x_1),...,F_p(x_p))$. Let a sample of size $n$ from the joint distribution $F(x_1,...,x_p)$ be denoted $\bm{x}_1,...,\bm{x}_n$ where the $j^{\text{th}}$ observation from this sample will be denoted $\bm{x}_j=(x_{1j},....,x_{pj})^\top$ for $j=1,...,n$. A sample from the underlying copula $C(u_1,...,u_p)$ will be denoted $\bm{u}_1....,\bm{u}_p$, where the $j^{\text{th}}$ observation from the sample will be denoted $\bm{u}_j=(u_{1j},....,u_{pj})^\top$. Further, let $U(0,1)$ denote the Uniform distribution with endpoints 0 and 1. The following algorithm will generate an approximate sample from the distribution $F(x_1,...,x_p)$. 
\begin{algorithm}[H]
\SetAlgoLined
\KwResult{Generating a sample from $F(x_1,...,x_p)$}
\For{$j=1$ to $n$}{
\For{$i=1$ to $p$}{
Simulate $u_{ij}^{(0)}$ from a $U(0,1)$ distribution\;
}
\For{$k=1$ to $m$}{
\For{$i=1$ to $p$}{
Generate $t$ from a $U(0,1)$ distribution;\\
Solve $C(u_{ij}^{(k)}|u_{1j}^{(k)},...,u_{(i-1)j}^{(k)},u_{(i+1)j}^{(k-1)},...,u_{pj}^{(k)})-t=0$ for $u_{ij}^{(k)}$\;
}
}
Set $\bm{u}_j=(u_{1j}^{(m)},...,u_{pj}^{(m)})^\top$\;
\For{$i=1$ to $p$}{
Solve $F_i(x_{ij})-u_{ij}=0$ for $x_{ij}$\;
}
Set $\bm{x}_j=(x_{1j},...,x_{pj})^\top$\;
}
\caption{Gibbs Sampler Applying Copula Theory}
\label{copgibbs}
\end{algorithm}
\[\]
Typically the number of iterations needs to be no more than 2 or 3 to provide a good approximate sample from $F(x_1,...,x_p)$.\\
One might wonder why it would be more advantageous to simulate from the copula and apply the probability integral transformation rather than simply using the actual distribution itself. There are two reasons for this:
\begin{enumerate}[(i)]
\item The partial derivatives of the copula can become quite long. If this is the case, applying the cdf will make it more difficult to code.
\item Many nonlinear solvers require both a compact domain and the function mapping into its domain. Many random variables (random vectors) will not satisfy these conditions. But the uniform distribution does.
\end{enumerate}
Even though it is the case that the copula does have a compact domain, the function might not be able to be evaluated numerically at points close to 0 (or possibly 1). For instance, in the case of the Gumbel copula, $\exp\left[-\left\{\sum_{i=1}^p(-\log(u_i))^\nu\right\}^{1/\nu}\right]$, $\nu\geq1$, $u_i\in[0,1]$ for $i=1,...,p$ values close to 0 might not be able to be evaluated numerically. If this is the case, typically choosing a lower bound for the solution slightly larger than 0 will typically work. If this is not amicable, the simply restarting the algorithm when this occurs is necessary.
\subsection{The Monte Carlo Gibbs sampler}
As seen before, the Gibbs sampler applying copula theory used the fact that 
\begin{enumerate}[(a)]
\item The underlying Copula is known
\item The partial derivatives can be evaluated for the underlying copula
\end{enumerate}
Although the partial derivatives can be evaluated with numeric differentiation, and hence can be evaluated as long as the underlying copula can be evaluated, this still may be problematic if the underlying copula cannot be identified. This would be the case if the cumulative distribution function is not known. If this is the case, a Monte Carlo integral may be required to approximate the underlying conditional cdf. After this is done, a Gibbs sampler can be employed using the approximated cdf.\\
Suppose that the random vector $(X_1,...,X_p)^\top$ has a joint pdf $f(x_1,...,x_p)$. Denote the joint pdf of $(X_1,...,X_{i+1},X_{i-1},...,X_p)^\top$ as $f(x_1,...,x_{i+1},x_{i-1},...,x_p)$. Denote the conditional pdf of $X_i|\{X_j=x_j\}_{j\neq i}$ as $f(x_i|x_1,...,x_{i-1},x_{i+1},...,x_p)$ and $f_i(x_i)$ as the marginal pdf of $X_i$ for $i=1,...,p$ then
\begin{align*}
 &F(x_i|x_1,...,x_{i-1},x_{i+1},...,x_p)\\
 =&\int_{-\infty}^{x_i}f(x|x_1,...,x_{i-1},x_{i+1},...,x_p)\,dx\\
 =&\mathbb{E}_{X_i}\left[\frac{f(x_1,...,X,...,x_p)}{f(X)f(x_1,...,x_{i-1},x_{i+1},...,x_p)}I(X\leq x_i)\right]
\end{align*}


In this case, by simulating $\{x_{il}\}_{l=1}^\ell$ from $f_i(x_i)$, the expectation can be approximated by the Monte Carlo integral, yielding

\begin{align}
 &F(x_i|x_1,...,x_{i-1},x_{i+1},...,x_p) \nonumber \\
 \approx& \frac{1}{\ell}\sum_{l=1}^\ell\frac{f(x_1,...,x_{il},...,x_p)}{f(x_{il})f(x_1,...,x_{i-1},x_{i+1},...,x_p)}I(x_{il}\leq x_i). \label{mccondit}
\end{align}
From (\ref{mccondit}), a Gibbs sampler can be developed based on the Monte Carlo conditional cdf. Let $(X_1,...,X_p)^\top$ have joint pdf $f(x_1,...,x_p)$ and marginal pdfs $f_i(x_i)$ for $i=1,...,p$. Define the conditional pdf of $X_i|\{X_j=x_j\}_{j\neq i}$ as $f(x_i|x_1,...,x_{i-1},x_{i+1},...,x_p)$. Denote a sample of observations from $f(x_1,...,x_p)$ to be $(\bm{x}_1,...,\bm{x}_n)$ where a single observation will be denoted $\bm{x}_j=(x_{1j},....,x_{pj})$, $j=1,...,n$. For a vector of values $\bm{y}_i=(y_{i1},...,y_{i\ell})$, such that $\bm{y}_i\in\text{Supp}(X_i)^\ell$, $i=1,...,p$, and an observation from the distribution $f(x_1,...,x_{i-1},x_{i+1},...,x_p)$, $(x_{1j},...,x_{(i-1)j},x_{(i+1)j},...,x_{pj})$, define
\[
\tilde{F}_i(x|x_{1j},...,x_{(i-1)j},x_{(i+1)j},...,x_p,\bm{y}_i)= \frac{1}{\ell}\sum_{l=1}^\ell\frac{f(x_{1j},...,y_{il},...,x_{pj})}{f\left(y_{il}\right)f\left(x_{1j},...,x_{(i-1)j},x_{(i+1)j},...,x_{pj}\right)}I(y_{il}\leq x).
\]
Then the following will provide an approximate sample from the distribution of $(X_1,...X_p)$
\begin{algorithm}[H]
\footnotesize
\KwResult{Generating an approximate sample from the distribution of $(X_1,...,X_p)^\top$}
\For{$i$ from 1 to $p$}
{
  \For{$l$ from 1 to $\ell$}
  {
    Simulate $y_{il}$ from $f_i(x_i)$\; 
   }
    Set $\bm{y}_{i}=(y_{i1},...,y_{i\ell})$
 }
 \For{$j$ from 1 to $n$}
 {
  \For{$i$ from 1 to $p$}
  {
   Simulate $x_{ij}^{(0)}$ from $f_i(x_i)$\\
   }
   
   \For{$k$ from 1 to $m$}
   {
   \For{$i$ from 1 to $p$}
   {
    Simulate $t$ from a $U(0,1)$ distribution\\
    \If{$t>\underset{x\in\{y_{i1},...,y_{i\ell}\}}{\max}\tilde{F}_i(x|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)$}
    {
    Set $x_{ij}^{(k)}=\underset{x\in\{y_{i1},...,y_{i\ell}\}}{\arg\max}\tilde{F}_i(x|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)$
    }
    \ElseIf{$t<\underset{x\in\{y_{i1},...,y_{i\ell}\}}{\min}\tilde{F}_i(x|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)$}
    {
    $x_{ij}^{(k)}=\underset{x\in\{y_{i1},...,y_{i\ell}\}}{\arg\min}\tilde{F}_i(x|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)$
    }
    \Else
    {
    Set $t^{\prime}$ to be $\underset{x\in\{y_{i1},...,y_{i\ell}\}}{\arg\max}\tilde{F}_i(x|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)-t$ constrained on $\tilde{F}_i(x|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)-t\leq0$\\
    Set $t^{\prime\prime}$ to be  $\underset{x\in\{y_{i1},...,y_{i\ell}\}}{\arg\min}\tilde{F}_i(x|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)-t$ constrained on $\tilde{F}_i(x|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)-t\geq0$\\
    Set 
    $x_{ij}^{(k)}=$
    \[
    \begin{cases} 
    t^\prime+\frac{(t^{\prime\prime}-t^\prime)(t  -\tilde{F}_i(t^{\prime}|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i))}{\tilde{F}_i(t^{\prime\prime}|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)-\tilde{F}_i(t^{\prime}|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)} & t^\prime\neq t^{\prime\prime}\\
    t^\prime & t^\prime=t^{\prime\prime}
    \end{cases}
    \]
    }
}
}
Set $\bm{x}_j=(x_{1j}^{(m)},...,x_{pj}^{(m)})$
}
\caption{Monte Carlo Gibbs sampler to simulate from $F(x_1,...,x_p)$}
\label{mccopgibbs}
\end{algorithm}

In order to find $\underset{x\in\{y_{i1},...,y_{i\ell}\}}{\arg\max}\tilde{F}_i(x|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)-t$ constrained on $\tilde{F}_i(x|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)-t\leq0$ and $\underset{x\in\{y_{i1},...,y_{i\ell}\}}{\arg\min}\tilde{F}_i(x|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)-t$ constrained on $\tilde{F}_i(x|x_{1j}^{(k)},...,x_{(i-1)j}^{(k)},x_{(i+1)j}^{(k-1)},...,x_p^{(k-1)},\bm{y}_i)-t\geq0$, a bisection method would typically by sufficient (numerical root finders would not work since the Monte Carlo integral will produce a step function, which is not continuous).\\
The inherent problem with this method is the speed at which this method can be done. Unlike the previous method which will give large samples virtually instantaneously, the Monte Carlo method can take minutes to provide one sample, if that sample is large enough. To ensure the speed is somewhat efficient, a smaller $\ell$ must be chosen and a smaller $m$ must be chosen. With a small $\ell$, however, accuracy is lost. Usually $\ell=n$ and $m=1$ will provide an adequate sample that can be used to perform inference. 
\subsection{Comparing the two methods}
In this section, the two methods will be compared using a bivariate distribution from (\ref{mvewcdf}). It will be analyzed by viewing a probability probability plot (p-p plot) of the empirical cdf of the data vs the true cdf of which the sample was supposed to come.
\subsubsection{The Gibbs Sampler from a multivariate distribution}
Suppose $\bm{X}=(X_1,...,X_p)^\top$ has joint distribution with cdf given by (\ref{mvewcdf}), then the Gibbs sample can be obtained by noting 
\[
C(u_1,...,u_p)=\exp\left[-\left\{\sum_{i=1}^p(-\log u_i)^\nu\right\}^{1/\nu}\right],
\]
and hence 
\[
C(u_i|u_{1},..,u_{i-1},u_{i+1},...,u_p)=\frac{\partial_{u_1}...\partial_{u_{i-1}}\partial_{u_{i+1}}...\partial_{u_p}C(u_1,...,u_p)}{\partial_{u_1}...\partial_{u_{i-1}}\partial_{u_{i+1}}...\partial_{u_p}C(u_1,...u_{i-1},1,u_{i+1},...,u_p)}, \]
$i=1,....,p$.\\
 $\partial_{u_1}...\partial_{u_{i-1}}\partial_{u_{i+1}}...\partial_{u_p}C(u_1,...u_{i-1},1,u_{i+1},...,u_p)$ is given in (\ref{genpdf}) with $R_i=-\log u_i$ and $r_i=\frac{1}{u_i}$. From the fact that $C(\cdot)$ is symmetric in it's arguments,  the indices of $u_1,...,u_p$ may be rearranged as follows. Let $(i_1,...,i_p)$ be a rearrangement of the $(1,....,p)$ such that $i_p$=$i$, $i=1,...,p$. Then 
 \begin{align}
 &\partial_{u_1}...\partial_{u_{i-1}}\partial_{u_{i+1}}...\partial_{u_p}C(u_1,...,u_p)\nonumber \\
 =&\partial_{u_{i_1}}...\partial_{u_{i_{p-1}}}C(u_{i_1},...,u_{i_p}) \nonumber \\ 
 =&\prod_{j=1}^{p-1}\left\{\frac{(-\log(u_{i_j}))^{\nu-1}}{u_{i_j}}\right\}\exp\left[-\left\{\sum_{j=1}^{p-1}(-\log{u_{i_j}})^\nu\right\}^{1/\nu}\right] \nonumber\\
 &\times\left\{\sum_{j=1}^{p-1}a_{j(p-1)}\left(\sum_{j=1}^p(-\log(u_{i_j}))^{\nu}\right)^{j(1-\nu)/\nu-(p-j-1)}\right\},
 \end{align}
 where $a_{jk}$ is defined in (\ref{genpdf}).\\
 In the bivariate case this is reduced to
 \begin{align}
 C(u_j|u_k)=\frac{(-\log u_k)^{\nu-1}}{u_k}\exp\left[-\left\{\sum_{i=1}^2(-\log u_i)^{\nu}\right\}^{1/\nu}\right]\left\{\sum_{i=1}^2(-\log u_i)^\nu\right\}^{\frac{\nu-1}{\nu}},
 \end{align}
 for $j,k=1,2$ $j\neq k$\\
 \subsubsection{The Monte Carlo Gibbs sampler}
 Thus using Algorithm \ref{copgibbs}, and a sample from the (\ref{mvewcdf}), to apply Algorithm \ref{mccopgibbs}, it can be seen that to obtain $f(x_1,x_2)$ from (\ref{mvewcdf}), (\ref{genpdf}) can be applied again, by letting $R_i(x_i)=-\log F(x_i|\lambda_i,\alpha_i,\theta_i)$ and $r_i(x_i)=\frac{f(x_i|\lambda_i,\alpha_i,\theta_i)}{F(x_i|\lambda_i,\alpha_i,\theta_i)}$ where $ F(x_i|\lambda_i,\alpha_i,\theta_i)$ is given in (\ref{ewcdf}) and $f(x_i|\lambda_i,\alpha_i,\theta_i)$ is given in (\ref{ewpdf}), $i=1,...,p$. An alternative approach would be to use $\bar{F}(y_i)=\exp(-y_i)I(y_i>0)$ and $f(y_i)=\exp(-y_i)I(y_i>0)$ applied to $\ref{genpdf}$ then use Algorithm 2 to obtain a sampler of size $n$ from $F(y_1,...,y_p)$. Then apply Theorem \ref{transform} to obtain a sample $F(x_1,...,x_p)$ as given in (\ref{mvewcdf}). 
 \subsubsection{Results}
 The following figures will display different samples simulated from (\ref{mvewcdf}) from both the Gibbs sampler using the copula and the Monte Carlo Gibbs sample. In each case, the Gibbs iteration size is $m=1$ and in each case and the sample size $n=1000$. For the Monte Carlo Gibbs sampler, a bin size is chosen to be $\ell=1000$. A runtime will be recorded in seconds for each of the samplers.
 \begin{figure}[H]
 \centering
 \begin{subfigure}{0.3\textwidth}
 \includegraphics[width=\textwidth]{plot2222}
 \caption{$(\lambda,\alpha,\theta,\nu)=(2,2,2,2)$\\
  MC runtime=12.84 s \\
  Copula runtime=0.26 s}
 \end{subfigure}
  \begin{subfigure}{0.3\textwidth}
  \includegraphics[width=\textwidth]{plot2522}
  \caption{$(\lambda,\alpha,\theta,\nu)=(2,.5,2,2)$\\
   MC runtime=13.1 s\\
    Copula runtime=0.25 s}
  \end{subfigure}
   \begin{subfigure}{0.3\textwidth}
   \includegraphics[width=\textwidth]{plot2553}
   \caption{$(\lambda,\alpha,\theta,\nu)=(2,.5,.5,3)$\\
   MC runtime=21.78 s\\
    Copula runtime=0.27 s}
   \end{subfigure}
 \end{figure} 
 It can be seen that the copula Gibbs sampler's runtime is far shorter than the Monte Carlo Gibbs sampler's runtime. In the case when the dependence parameter is increase from 2 to 3, the Monte Carlo based Gibbs sampler has its runtime almost doubled, where as the Copula based Gibbs sampler remains unaffected. It can almost be seen that the copula based Gibbs sampler provides tighter plots (the PP plot is almost a straight line).
 \subsection{Other Sampling methods}
  Another sampling method that may used is a sampler based on the Metropolis Hastings algorithm. Although the few assumptions may make this sampler appealing, it is not recommended to use this sampler when trying to sample from a multivariate distribution as its convergence is slow. This will provide a sampler that has a very slow runtime and will often provide inadequate samples. 
\section{Inferential Procedures}
In this section, methods of obtaining confidence regions will be established by using asymptotic results and computationally intensive procedures. In chapter 4, these methods will be compared and evaluated based on their estimated confidence coefficient.
\subsection{Estimation}
In most situations of parametric modeling, the maximum likelihood estimators (MLEs) are typically the most widely used point estimator as they tend to have nice properties (such as consistency, asymptotic efficiency, and asymptotic normality). In this subsection, methods of obtaining the maximum likelihood estimators will be established.\\ 
Before methods to obtain MLEs are given, some notation will be necessary.
Let $(\bm{X}_1,...,\bm{X}_n)$ be a random sample of size $n$ from a distribution with pdf $f(x_1,...,x_p|\bm{\theta})$ with $\bm{X}_j=(X_{1j},...,X_{pj})^\top$ for $j=1,...,n$ and $\bm{\theta}=(\theta_1,...,\theta_d)^\top\in\Theta\subset\mathbb{R}^d$. The likelihood function will denoted
\begin{align}
L(\bm{\theta}|\bm{X}_1,...,\bm{X}_n)=\prod_{j=1}^nf(X_{1j},...,X_{pj}|\bm{\theta})
\label{likelihood}.
\end{align}
In many situations, the log-likelihood function is considered. The log-likelihood function will be denoted by
\begin{align}
\ell(\bm{\theta}|\bm{X}_1,...,\bm{X}_n)=\sum_{j=1}^n\log f(X_{1j},...,X_{pj}|\bm{\theta}).
\end{align}
The maximum likelihood estimator for $\bm{\theta}$ is
\begin{align}
\hat{\bm{\theta}}_{MLE}=\underset{\bm{\theta}\in\Theta}{\arg\max}\,L(\bm{\theta}|\bm{X}_1,...,\bm{X}_n)=\underset{\bm{\theta}\in\Theta}{\arg\max}\,\ell(\bm{\theta}|\bm{X}_1,...,\bm{X}_n).
\end{align}
In the case where the likelihood function is differentiable the MLEs can be found by solving the system of equations
\begin{align*}
\frac{\partial \ell(\bm{\theta}|\bm{X}_1,...,\bm{X}_n)}{\partial\theta_i}&=0,&i=1,...,d.
\end{align*}
If this is not feasible, direct maximization is necessary.
\subsection{Asymptotic Theory}
In most situations, it is of interest to not only obtain estimates of the parameters, but also to obtain the distribution of those estimators. Typically, outside of simple cases, this is not feasible. Thus it is of interest to find the approximate distribution of the MLEs, and then use that approximate distribution to perform inference.\\
First, it is of interest to more rigorously define what an approximate distribution means. 
Let $\bm{T}_n:\text{Supp}(\bm{X})^n\rightarrow \mathbb{R}^d$ be a statistic of a random sample of size $n$, $(\bm{X}_1,...,\bm{X}_n)$, from the distribution $F(x_1,...,x_p)$. Let $G_n(\bm{t})$ denoted the cdf ${T}_n$. Let $\bm{T}$ be a random vector on $\mathbb{R}^d$ with cdf $G(\bm{t})$. ${T}_n$ is said to converge in distribution to $\bm{T}$ if 
\begin{align*}
G_n(\bm{t})\rightarrow G(\bm{t})&\text{ point wise},&\text{as }n\rightarrow\infty.
\end{align*}
$\bm{T}_n$ converging in distribution to $\bm{T}$ will typically be denoted $\bm{T}_n\overset{d}{\rightarrow}\bm{T}$ as $n\rightarrow \infty$. In many cases, it will be convenient to write, $\bm{T}_n\overset{d}{\rightarrow}G(\bm{t})$ as $n\rightarrow \infty$ to denote $\bm{T}_n$ converges in distribution to a random vector $\bm{T}$ with distribution $F(\bm{t})$.\\
Using this fact, the distribution of the MLEs can be approximated as follows. Under suitable regularity conditions
\begin{align}
 &\sqrt{n}(\hat{\bm{\theta}}_{MLE}-\bm{\theta})\overset{d}{\rightarrow}N_d(\bm{0},\mathscr{I}(\bm{\theta})^{-1}), & \textrm{as }n\rightarrow\infty
 \label{asymnorm}
\end{align}
where $N_d(\bm{\mu},\Sigma)$ denotes the normal distribution with mean vector $\bm{\mu}\in\mathbb{R}^{d}$ and symmetric positive definite matrix $\Sigma\in\mathbb{R}^{d\times d}$ and
\begin{align*}
\mathscr{I}(\bm{\theta})_{ij}&=\mathbb{E}\left(\frac{\partial}{\partial\theta_i}\log f(\bm{X};\bm{\theta})\right)\left(\frac{\partial}{\partial\theta_j}\log f(\bm{X};\bm{\theta})\right),&i,j=1,...,d.
\end{align*}

\noindent is called the \emph{information matrix}. Since the true information matrix is not known, it can be estimated by
\begin{align*}
\widehat{\mathscr{I}(\bm{\theta})}&=\mathscr{I}(\hat{\bm{\theta}}_{MLE}).
\end{align*} 

In many cases, the information matrix is difficult to compute and thus it is of interest to use the observed information matrix. Suppose we have a random sample $(\bm{X}_1,...,\bm{X}_n)$ from the distribution with pdf $f(x_1,...,x_p|\bm{\theta})$, where $\bm{\theta}=(\theta_1,...,\theta_d)^\top\in\Theta\subset\mathbb{R}^d$. The $ij^{th}$ entry of the observed information matrix is defined to be
\begin{align}
I(\bm{\hat{\theta}}_{MLE})_{ij}&=-\frac{\partial^2}{\partial\theta_i\partial\theta_j}\ell(\bm{\theta}|\bm{X}_1,...,\bm{X}_n)_{\bm{\theta}=\hat{\bm{\theta}}_{MLE}},&i,j=1,...,d.
\end{align}
Again, under suitable regularity conditions,
\begin{align}
\frac{1}{n}I(\hat{\bm{\theta}}_{MLE})&\overset{p}{\rightarrow}\mathscr{I}(\bm{\theta}),&\text{as } n\rightarrow\infty.\\
I(\hat{\bm{\theta}}_{MLE})^{1/2}(\hat{\bm{\theta}}_{MLE}-\bm{\theta})&\overset{d}{\rightarrow} N_d(\bm{0},J),&\text{as }n\rightarrow\infty. \label{asymptot}
\end{align}
where $J$ is the $d\times d$ identity matrix.
\subsubsection{Assessing Asymptotic Normality for the MVEW distribution}
In this section, the asymptotic normality will be assessed for the bivariate $\text{MVEW}(\bm{\lambda},\bm{\alpha},\bm{\theta},\nu)$ model defined in (\ref{mvewcdf}) for the maximum likelihood estimators under select cases of the values of the ``true parameters.'' In each situation, 250 samples of size 50, 100 and 250 were obtained using the copula Gibbs sampler with an iteration of 1 and maximum likelihood estimators were obtained for each sample using the \texttt{optim} function in R. The following plots show histograms for the MLEs in each case to depict the convergence to normality. For each situation, it will be assumed that each marginal distribution is the same for simplicity (\emph{i.e.} $\lambda_1=\lambda_2$, $\alpha_1=\alpha_2$, and $\theta_1=\theta_2$).

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm5551}
\caption{Histograms for the MLEs when $\lambda=0.5$, $\alpha=0.5$, $\theta=0.5$, and $\nu=1$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm5553}
\caption{Histograms for the MLEs when $\lambda=0.5$, $\alpha=0.5$, $\theta=0.5$, and $\nu=3$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm3551}
\caption{Histograms for the MLEs when $\lambda=3$, $\alpha=0.5$, $\theta=0.5$, and $\nu=1$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm3553}
\caption{Histograms for the MLEs when $\lambda=3$, $\alpha=0.5$, $\theta=0.5$, and $\nu=3$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm5351}
\caption{Histograms for the MLEs when $\lambda=0.5$, $\alpha=3$, $\theta=0.5$, and $\nu=1$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm5353}
\caption{Histograms for the MLEs when $\lambda=0.5$, $\alpha=3$, $\theta=0.5$, and $\nu=3$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm3351}
\caption{Histograms for the MLEs when $\lambda=3$, $\alpha=3$, $\theta=0.5$, and $\nu=1$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm3353}
\caption{Histograms for the MLEs when $\lambda=3$, $\alpha=3$, $\theta=0.5$, and $\nu=3$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm5531}
\caption{Histograms for the MLEs when $\lambda=0.5$, $\alpha=0.5$, $\theta=3$, and $\nu=1$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm5533}
\caption{Histograms for the MLEs when $\lambda=0.5$, $\alpha=0.5$, $\theta=3$, and $\nu=3$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}


\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm3531}
\caption{Histograms for the MLEs when $\lambda=3$, $\alpha=0.5$, $\theta=3$, and $\nu=1$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm3533}
\caption{Histograms for the MLEs when $\lambda=3$, $\alpha=0.5$, $\theta=3$, and $\nu=3$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\ContinuedFloat
\includegraphics[scale=0.8]{anorm5331}
\caption{Histograms for the MLEs when $\lambda=0.5$, $\alpha=3$, $\theta=3$, and $\nu=1$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm5333}
\caption{Histograms for the MLEs when $\lambda=0.5$, $\alpha=3$, $\theta=3$, and $\nu=3$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm3331}
\caption{Histograms for the MLEs when $\lambda=3$, $\alpha=3$, $\theta=3$, and $\nu=1$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{anorm3333}
\caption{Histograms for the MLEs when $\lambda=3$, $\alpha=3$, $\theta=3$, and $\nu=3$\\
Where $n=50$ (first row), $n=100$ (second row), and $n=250$ (third row)}
\end{figure}

From these histograms, it can be clearly seen that if the true value of $\nu$ is 1, then the maximum likelihood estimates of $\nu$ seem to never converge to a normal distribution. This should make sense, since the space for $\nu$ is $\{\nu:\nu\geq1\}$ is a closed set. If $\nu$ is chosen to be on the boundary of $\{\nu:\nu\geq1\}$ the asymptotics should not work. In the case when the true value of $\nu$ is much larger than 1, the convergence to normality seems to be quick. Even when the sample size is 50, the distribution of the MLE for $\nu$, when the ``true value'' of $\nu$ is large seems to be fairly symmetric and mound shaped. Another thing to note is that the distribution of the MLE for $\nu$ seems to be independent of the values of the underlying parameters in the marginal distributions (at least there is no recognizable pattern).\\

For the parameter $\theta$, it can be seen that when the true value of $\theta$ is large, the convergence to normality for the MLEs of $\theta$ tends to be a bit slower than when the value of $\theta$ is small. In the case when $\theta$ is large, there seems to be a  convergence to normality for the MLE of $\theta$ outside of a few outlying values, which introduce a slight skew to the distribution of $\theta$. \\

There seems to be an interrelation between the distribution of the MLEs of $\alpha$ and the distribution of the MLEs of $\theta$. When $\theta$ is large, the MLEs for $\alpha$ tend to converge very quickly to normality. In all cases shown, a sample of size 50 is sufficient. If $\theta$ is small, however, the speed of convergence to normality for the MLEs of $\alpha$ tends to be much slower. In the case when the true value of $\alpha$ is small, a sample of size 100 or larger is recommended to conclude convergence to normality of the MLEs for $\alpha$. If the true value of $\alpha$ is large, samples should be at least of size 250 to conclude convergence to normality for the MLEs of $\alpha$. The reason for this is small values of $\theta$ tend to introduce a positive skew on the distribution of the MLEs for $\alpha$ \\

The parameters $\alpha$ and $\lambda$ are also interrelated. If the ``true value'' of $\alpha$ is small, then in many cases, the convergence of of the MLEs of $\lambda$ is very slow. In many cases, it seems that even a sample of size 250 may not be sufficient. This speed seems to be slowed down even further if the ``true value'' of $\theta$ is large. It should be noted, that lack of fit to normality seems to be the result of a few outliers. This may be the result of using numeric optimization. Further investigation into different optimization producers should be done before any firm conclusions are drawn.\\

If the ``true value'' of $\alpha$ is large, however, the MLEs for $\lambda$ seem to converge at a moderate pace. In all cases, a sample of size 100 seems sufficient to conclude that the asymptotics has ``kicked in''.     

\subsubsection{Asymptotic Inference}
 It is typically of interest to perform inference on the parameters of a distribution. In this section, for $\bm{\theta}=(\theta_1,...,\theta_d)^\top$, procedures to test $H_0:\,\bm{\theta}=\bm{\theta}_0$ will be introduced, and then inversion of these tests will provide methods for obtaining asymptotic confidence regions. Two procedures will be investigated: a test based on a Mahalanobis distance and a test based on the likelihood ratio test statistic. Before this is discussed, notation will be discussed again. Let $(\bm{X}_1,...,\bm{X}_n)$ be a random sample from the distribution $f(x_1,...,x_p|\bm{\theta})$, where $\bm{\theta}=(\theta_1,...,\theta_d)^\top\in\Theta\subset\mathbb{R}^d$.  
 \subsubsection{A test based on a Mahalanobis distance}
 From (\ref{asymptot}), it can easily be seen by applying the continuous mapping theorem that 
 \begin{align}
 (\hat{\bm{\theta}}_{MLE}-\bm{\theta})^\top I(\hat{\bm{\theta}}_{MLE})(\hat{\bm{\theta}}_{MLE}-\bm{\theta})&\overset{d}{\rightarrow} \chi^2_{(d)},&\text{ as }n\rightarrow\infty,  \label{mdist}
 \end{align}
 where $\chi^2_{(d)}$ denotes the chi squared distribution with $d$ degrees of freedom. \\
 Suppose it is of interest to test $H_0:\bm{\theta}=\bm{\theta}_0\in\Theta$. Using (\ref{mdist}), a size $\alpha$ test can be constructed by using the rejection region
 \begin{align}
 \{(\bm{x}_1,...,\bm{x}_n)\in\text{Supp}(\bm{X})^n:(\hat{\bm{\theta}}_{MLE}-\bm{\theta}_0)^\top I(\hat{\bm{\theta}}_{MLE})(\hat{\bm{\theta}}_{MLE}-\bm{\theta}_0)\geq \chi^2_{(d),\alpha}\},
 \end{align}
 in which $\chi^2_{(d),\alpha}$ denotes the $\alpha$th critical value of the chi square distribution with $d$ degrees of freedom.\\
 This test can be inverted to provide asymptotic confidence regions, where the $100(1-\alpha)\%$ confidence region for $\bm{\theta}$ is given by
 \begin{align}
 \{\bm{\theta}\in\Theta:(\hat{\bm{\theta}}_{MLE}-\bm{\theta})^\top I(\hat{\bm{\theta}}_{MLE})(\hat{\bm{\theta}}_{MLE}-\bm{\theta})\leq \chi^2_{(d),\alpha}\}.
 \label{infoconf}
 \end{align}
 \subsubsection{Likelihood Ratio Procedures}
 Another method of performing a test using the asymptotic distribution of the MLEs is by using a likelihood ratio test. Again, suppose $(\bm{X}_1,...,\bm{X}_n)$, where $\bm{X}_j=(X_{1j},...,X_{pj})^\top$ for $j=1,...,n$,  is a random sample from $f(x_1,...,x_P|\bm{\theta})$, where $\bm{\theta}\in\Theta\subset\mathbb{R}^d$.
 The \textbf{likelihood ratio test statistic} is defined to be
 \begin{align}
 \Lambda(\bm{\theta}_0|\bm{X}_1,...,\bm{X}_n)=2\ell(\hat{\bm{\theta}}_{MLE}|\bm{X}_1,...,\bm{X}_n)-2\ell(\bm{\theta}_0|\bm{X}_1,...,{X}_n).
 \label{LRteststat}
 \end{align} 
 \indent Suppose (\ref{asymnorm}) holds. Under the assumption $\bm{\theta}=\bm{\theta}_0$ and additional regularity conditions
 \begin{align}
 \Lambda(\bm{\theta}_0|\bm{X}_1,...,\bm{X}_n)&\overset{d}{\rightarrow}\chi^2_{(d)},&\text{ as } n\rightarrow\infty,
 \label{LRasymptotics}
 \end{align} 
 where $\chi^2_d$ denotes the chi square distribution with $d$ degrees of freedom.\\
 Under this situation, an asymptotic size $\alpha$ test can be constructed to test $H_0:\,\bm{\theta}=\bm{\theta}_0$ by using the rejection region
 \begin{align}
 \{(\bm{x}_1,...\bm{x}_n)\in\text{Supp}(\bm{X})^n:\Lambda(\bm{\theta}_0|\bm{x}_1,...,\bm{x}_n)\geq \chi^2_{(d),\alpha}\},
 \end{align}
 where $\chi^2_{(d),\alpha}$ denotes the $\alpha$th critical value for a $\chi^2_{(d)}$ distribution. \\
 Inverting this test will allow a method for obtaining confidence regions. Suppose $(\bm{x}_1,...,\bm{x}_n)$ is an observed sample of size $n$ from $f(x_1,...,x_p|\bm{\theta})$, then an asymptotic $100(1-\alpha)\%$ confidence region for $\theta$ can be obtained by considering
 \begin{align}
 \{\bm{\theta}\in\Theta:\Lambda(\bm{\theta}|\bm{x}_1,...,\bm{x}_n)\leq \chi^2_{(d),\alpha}\}.
 \label{LRconf}
 \end{align}
 \section{Bootstrap Based Procedures}
 In this section, parametric bootstrapping methods will be discussed for obtaining confidence regions for a collection of parameters of interest. To achieve this, the bootstrap methodology will be discussed  (in a parametric model) and then methods of estimating the covariance matrix will be examined. Methods of assessing the validity of this will be discussed in chapter 4.\\
 \subsection{The Bootstrap Methodology}
 For a random sample of size $n$, $(\bm{X}_1,...,\bm{X}_n)$, from a distribution given by $F(\bm{x}|\bm{\theta})$, where $\bm{X}_i\in\text{Supp}(\bm{X})\subset\mathbb{R}^p$, for $i=1,...,n$ $\bm{\theta}\in\Theta\subset\mathbb{R}^d$, let $\hat{\bm{\theta}}:\text{Supp}(\bm{X})^n\rightarrow\Theta$ be an estimator of $\bm{\theta}$. \\
Then, for an observed sample of $(\bm{x}_1,....,\bm{x}_n)$, $\bm{\theta}$ can be estimated by $\hat{\bm{\theta}}=\hat{\bm{\theta}}(\bm{x}_1,...,\bm{x}_n)$. From  this, $B$ samples of size $n$ can be simulated from $F(\bm{x}|\hat{\bm{\theta}})$, $(\bm{x}^*_{1,1},...,\bm{x}^*_{n,1}),...,(\bm{x}^*_{1,B},...,\bm{x}^*_{n,B})$. For each $b=1,...B$, let $\hat{\bm{\theta}}^*_b=\hat{\bm{\theta}}(\bm{x}^*_{1,b},....,\bm{x}^*_{n,b})$. Then, $(\hat{\bm{\theta}}^*_1,....,\hat{\bm{\theta}}^*_B)$ is a random sample from the distribution of the random vector $\hat{\bm{\theta}}(\bm{X}^*_1,...,\bm{X}^*_n)$, where $\bm{X}^*_1,...,\bm{X}^*_n\overset{i.i.d.}{\sim}F(\bm{x}|\hat{\bm{\theta}})$.\\
With this information in hand, two things can be obtained:
\begin{enumerate}[(i)]
\item Estimates of the covariance matrix of $\hat{\bm{\theta}}$, and
\item Confidence regions for $\bm{\theta}$.
\end{enumerate}
\subsection{Estimating the covariance matrix of $\hat{\bm{\theta}}$}
As seen before, it is of interest to estimate the covariance matrix for $\hat{\bm{\theta}}=\hat{\bm{\theta}}(\bm{X}_1,...,\bm{X}_n)$, where $\hat{\bm{\theta}}:\text{Supp}(\bm{X})^n\rightarrow\Theta\subset\mathbb{R}^d$, and $\bm{X}_1,....,\bm{X}_n\overset{i.i.d.}{\sim}F(\bm{x}|\theta)$. It should be noted that the true covariance matrix for $\hat{\bm{\theta}}$, denoted $\Sigma_{\hat{\bm{\theta}}}$, is given by
\begin{align}
\Sigma_{\hat{\bm{\theta}}}&=\mathbb{E} \left[ (\hat{\bm{\theta}}-\mathbb{E}[\hat{\bm{\theta}}])(\hat{\bm{\theta}}-\mathbb{E}[\hat{\bm{\theta}}])^\top \right].
\label{truecovariance}
\end{align}
   Using the notation from \S3.3.1, an estimator for $\hat{\bm{\theta}}$ can be established by defining
\begin{align}
\bar{\hat{\bm{\theta}}}^*&=\frac{1}{B}\sum_{b=1}^B\hat{\bm{\theta}}^*_b,\\
\hat{\Sigma}^*_{\hat{\bm{\theta}}}&=\frac{1}{B-1}\sum_{b=1}^B(\hat{\bm{\theta}}^*_b-\bar{\hat{\bm{\theta}}}^*)(\hat{\bm{\theta}}^*_b-\bar{\hat{\bm{\theta}}}^*)^\top.
\label{bscovariance}
\end{align}
In this case, $\hat{\Sigma}^*_{\hat{\bm{\theta}}}$ should be a consistent estimator of $\Sigma_{\hat{\bm{\theta}}}$. If this is the case, and $\hat{\bm{\theta}}$ is a consistent estimator of $\bm{\theta}$, then under additional regularity conditions
\begin{align}
\hat{\Sigma}^{*-1/2}_{\hat{\bm{\theta}}}(\hat{\bm{\theta}}-\bm{\theta})\overset{d}{\rightarrow}N_d(\bm{0},J)&\text{ as }n\rightarrow\infty, \label{bsasymptotic}
\end{align}
where $J$ is the $d\times d$ identity matrix.\\
From this, it can be seen that 
\begin{align}
(\hat{\bm{\theta}}-\bm{\theta})^\top\hat{\Sigma}^{*-1}_{\hat{\bm{\theta}}}(\hat{\bm{\theta}}-\bm{\theta})\overset{d}{\rightarrow}\chi^2_{(d)}&\text{ as }n\rightarrow\infty.\label{bschisq}
\end{align}
\subsection{Inferential Procedures Using the Bootstrap Estimator for the Covariance Matrix}
Suppose $\bm{X}_1,...,\bm{X}_n\overset{i.i.d.}{\sim}F(\bm{x}|\bm{\theta})$, where $\bm{\theta}\in\Theta\subset\mathbb{R}^d$. In this case, $\bm{\theta}$ can be estimated with $\hat{\bm{\theta}}(\bm{X}_1,...,\bm{X}_n)$, where $\hat{\bm{\theta}}:\text{Supp}(\bm{X})^n\rightarrow\Theta\subset\mathbb{R}^d$ is a consistent estimator of $\bm{\theta}$. Suppose further that $\Sigma_{\hat{\bm{\theta}}}$, as defined in (\ref{truecovariance}), is estimated with $\hat{\Sigma}^*_{\hat{\bm{\theta}}}$ as defined in (\ref{bscovariance}). From (\ref{bschisq}), an $\alpha$-level test can be constructed to test $H_0:\bm{\theta}=\bm{\theta}_0$, by using the rejection region
\begin{align}
\{\bm{x}_1,...,\bm{x}_n\in\text{Supp}(X)^n:(\hat{\bm{\theta}}-\bm{\theta})^\top\hat{\Sigma}^{*-1}_{\hat{\bm{\theta}}}(\hat{\bm{\theta}}-\bm{\theta})\geq\chi^2_{(d)}\}
\label{bschisqtest}
\end{align}
If $\bm{X}_1,...,\bm{X}_n$ is observed to be $\bm{x}_1,...,\bm{x}_n$, (\ref{bschisqtest}) can be inverted to yield a $100(1-\alpha)\%$ confidence region for $\bm{\theta}$ as 
\begin{align}
\{\bm{\theta}\in\Theta\subset\mathbb{R}^d:(\hat{\bm{\theta}}-\bm{\theta})^\top\hat{\Sigma}^{*-1}_{\hat{\bm{\theta}}}(\hat{\bm{\theta}}-\bm{\theta})\leq\chi^2_{(d)}\}. \label{bsconfreg}
\end{align}
\subsection{Other Bootstrap Based Procedures}
Efron and Tibshirani (1993) note ``One of the principal goals of bootstrap is to produce good confidence intervals automatically. `Good' means that the bootstrap intervals should closely match exact confidence intervals in those special situations where statistical theory yields an exact answer, and should give dependably accurate coverage probabilities in all situations.'' Because of this it is of interest to determine if the confidence regions procedure in (\ref{bsconfreg}) provide good confidence regions, or if there is a need for \emph{bias correction}.\\
To assess if there is need for bias correction for the asymptotic procedure in (\ref{bsconfreg}), the analogous bootstrap z-interval procedure will be compared with the $\text{BC}_a$ interval by comparing the length of each confidence interval and the coverage probabilities. Hence, these procedures will be discussed briefly.\\
\subsubsection{The Bootstrap z-Interval} 
Suppose $\bm{X}_1,...,\bm{X}_n\overset{i.i.d.}{\sim}F(\bm{x}|\bm{\theta})$, where $\bm{\theta}=(\theta_1,...,\theta_d)\in\Theta\subset\mathbb{R}^d$, where $\Theta=\Theta_1\times\Theta_2\times...\times\Theta_d$. Let $\hat{\bm{\theta}}=\hat{\bm{\theta}}(\bm{X}_1,...,\bm{X}_n)$ be a consistent estimator of $\bm{\theta}$, and let $\hat{\sigma}^*_{\hat{\theta}_i}$ denote the square root of the $ii^{\text{th}}$ entry of $\hat{\Sigma}^*_{\hat{\bm{\theta}}}$, $i=1,...,d$, where $\hat{\Sigma}^*_{\hat{\bm{\theta}}}$ is defined in (\ref{bscovariance}). From \ref{bsasymptotic}, it can be seen that a  confidence interval for $\theta_i$ with coverage $(1-\alpha)$ can be constructed as
\begin{align}
\{\theta_i\in\Theta_i:\hat{\theta}_i-z_{\alpha/2}\cdot\hat{\sigma}^*_{\hat{\theta}_i}\leq\theta_i\leq\hat{\theta}_i+z_{\alpha/2}\cdot\hat{\sigma}^*_{\hat{\theta}_i}\},
\end{align} 
where $i=1,...,d$ and $z_{\alpha/2}$ denotes the $\alpha/2^{\text{th}}$ critical value for the standard normal distribution. 
\subsubsection{The Bootstrap $\text{BC}_a$ Interval}
The bootstrap $\text{BC}_a$ (which stands for bias-corrected and accelerated) interval can be seen as an alternative to the bootstrap percentile method, discussed in Efron and Tibshirani (1993), in which the percentiles are dependent on $\hat{z}_0$, called the \emph{bias-correction}, and $\hat{\alpha}$, called the \emph{ }. Efron and Tibshirani note ``Roughly speaking, $\hat{z}_0$ measures the median bias of $\hat{\theta}^*$, that is, the discrepancy between the median of $\hat{\theta}^*$ and $\hat{\theta}$... The quantity $\hat{\alpha}$ is called the \emph{acceleration} because it refers to the rate of change of the standard error of $\hat{\theta}$ with respect to the true parameter value $\theta$.'' The algorithm for computing the $\text{BC}_a$ interval can be found in Efron and Tibshirani (1993). Efron and Tibshirani note ``Roughly speaking, $\hat{z}_0$ measures the median bias of $\hat{\theta}^*$, that is, the discrepancy between the median of $\hat{\theta}^*$ and $\hat{\theta}$... The quantity $\hat{\alpha}$ is called the \emph{acceleration} because it refers to the rate of change of the standard error of $\hat{\theta}$ with respect to the true parameter value $\theta$.'' The algorithm for computing the $\text{BC}_a$ interval can be found in Efron and Tibshirani (1993). 


\chapter{Simulation Study and Results}
In this chapter, a simulation study is conducted in order to assess the results of this thesis. Samples were obtained from the bivariate MVEW model discussed in (\ref{mvewcdf}) using a copula Gibbs sampler, in which the marginal distributions were assumed equal. For each combination of values of the parameters, $\lambda=0.5,1,3$, $\alpha=0.5,1,3$, $\theta=0.5,1,3$, and $\nu=1,2,3$, 250 samples of sizes $n=50,100$, and $250$ were obtained. \\
For each sample, the MLEs were computed using the Nelder-Mead method from the \texttt{optim} function in R. For each combination of parameters the mean squared error (MSE) and bias were estimated using Monte Carlo integrals to assess the accuracy of these estimates.\\
For each sample's MLEs, the observed information matrix was computed, and then the Mahalanobis distance was computed by using (\ref{mdist})
for the model in (\ref{mvewcdf}), where $\bm{\theta}_0$ are the true values of the parameters \emph{i.e.} the values of the parameters from (\ref{mvewcdf}) of which the sample was sampled. Coverage probabilities for this method were estimated by finding the proportion of samples that satisfied (\ref{infoconf}).\\
Similarly a bootstrap based Mahalanobis distance was obtained for each sample's MLEs using the covariance matrix found in (\ref{bscovariance}) and a Mahalanobis distance was computed using (\ref{bschisq}). Coverage probabilities were estimated using (\ref{bsconfreg}). \\
A likelihood ratio test was also performed using (\ref{LRasymptotics}), in which coverage probabilities were estimated by observing the proportion of samples that satisfied (\ref{LRconf}).\\
To assess if there is a necessity for bias correction, a bootstrap z-interval is provided along with a bootstrap $\text{BC}_a$ as well. Different collections of values are chosen for the parameters. For each collection of the parameters, and each interval type, the average length and coverage probabilities are given. Different collections of values are chosen for the parameters. For each collection of the parameters, and each interval type, the average length and coverage probabilities are given.
\section{Bias and MSE}
In this section, the bias and MSE of each parameter will be examined. From this, unbiasedness and consistency will also be examined. The effects of values of the parameters on the bias and MSE will also be examined, as these effects are not negligible. \\
\subsection{Consistency}
As seen in table \ref{biasmse50}, \ref{biasmse100}, and \ref{biasmse250} in the appendix of tables, the MSE is decreasing. In most cases, it can be seen that the MSE is decreasing by $\frac{1}{n}$. Since this is the case, it should be inferred that the estimators are consistent. Also it can be noted that the bias is decreasing in the same fashion. In the case of $\nu>1$, the MLE of $\nu$ seems to be unbiased since the estimated bias seems to fluctuate between being positive and negative. Of course, when $\nu=1$, the MLE of $\nu$ is biased, due to $\nu=1$ is on the boundary of the parameter space.
\subsection{Effect of the Parameter Values on the Bias and MSE}
In all cases where $\alpha=0.5$, it can be seen that the MSE of $\hat{\lambda}_{MLE}$ is large. In some cases, an unusually large MSE can be seen. This is especially true when $n=50$, $\lambda=3$, and $\alpha=0.5$. A big cause of this can be attributed to numerical error. This numerical error is likely due to a flat likelihood function. Figure 4.1 provides and example. Some examples can be seen in Tables 4.1 and 4.2. A complete list of tables can be found in A.1.
\begin{figure}[H]
\includegraphics[scale=0.5]{LambdaVsLikelihood}
\caption{Lambda vs. the likelihood function for a particular sample}
\label{lvl}
\end{figure}
 It can also be seen that when $\theta=3$, it also has an unusually high variability. This can probably be attributed to a simply a large variability in $\theta$ when $\theta$ is large. The parameter $\nu$ seems to have no affect on the bias and MSE of the other parameters.

\subsection{Some Tabular Examples}
The following tables provides estimates of the bias (second column) and MSE (third column) for each parameter of the MVEW distribution in (\ref{mvewcdf}). The parameter in question is provided (first column) and the value from which the distribution was simulated is provided (fourth column). Sample sizes for each set are provided as well (first row).
 \begin{table}[H]
 \centering
\begin{tabular}{|c|r|r|c|}
\hline
\multicolumn{4}{|c|}{n=50}\\ \hline
Parameter & Bias & MSE & True Value \\ \hline
$\lambda$ &   0.007 &   0.071 &   0.5  \\ \hline
 $\alpha$ &  0.315 &   1.277 &   3.0  \\ \hline  
 $\theta$ &   0.028 &   0.230 &   0.5  \\ \hline  
  $\nu$ &    0.031 &   0.056 &   1.0\\ \hline \hline
$\lambda$ &   0.023 &   0.418 &   3.0  \\ \hline
 $\alpha$ &  0.369 &   1.297 &   3.0  \\ \hline
 $\theta$ &   0.012 &   0.215 &   0.5  \\ \hline
  $\nu$ &     0.026 &   0.052 &   1.0\\ \hline \hline
  $\lambda$ &   0.735 &   4.349 &   0.5  \\ \hline
 $\alpha$ &  0.027 &   0.142 &   0.5  \\ \hline 
 $\theta$ &   0.627 &   3.293 &   3.0  \\ \hline
  $\nu$ &    0.030 &   0.054 &   1.0\\ \hline \hline
 $\lambda$ &   1.463 &  10.350 &   3.0  \\ \hline
 $\alpha$ &  0.028 &   0.136 &   0.5  \\ \hline
 $\theta$ &   0.295 &   1.997 &   3.0  \\ \hline
  $\nu$ &    0.027 &   0.052 &   1.0\\ \hline
 \end{tabular} 
 \caption{Bias MSE Example for $n=50$}
 \end{table}
\begin{table}[H]
\centering
\begin{tabular}{|c|r|r|c|}
\hline
\multicolumn{4}{|c|}{n=250}\\ \hline
Parameter & Bias & MSE & True Value \\ \hline
 $\lambda$ &   0.023 &   0.117 &   0.5  \\ \hline
 $\alpha$ &   0.404 &   1.582 &   3.0  \\ \hline
 $\theta$ &     0.059 &   0.340 &   0.5  \\ \hline
  $\nu$ &     $-$0.028 &   0.433 &   3.0\\ \hline \hline
  $\lambda$ &   0.026 &   0.337 &   3.0 \\ \hline
$\alpha$ &   0.219 &   0.788 &   3.0 \\ \hline
$\theta$ &     0.000 &   0.157 &   0.5 \\ \hline
 $\nu$ &      $-$0.003 &   0.307 &   3.0 \\ \hline\hline
  $\lambda$ &   0.823 &   3.521 &   0.5  \\ \hline
 $\alpha$ &   0.026 &   0.146 &   0.5  \\ \hline
 $\theta$ &    1.019 &   4.364 &   3.0  \\ \hline
  $\nu$ &    $-$0.055 &   0.444 &   3.0\\ \hline \hline
   $\lambda$ &   5.123 &  44.064 &   3.0  \\ \hline
 $\alpha$ &   0.039 &   0.171 &   0.5  \\ \hline
 $\theta$ &    1.047 &   4.432 &   3.0  \\ \hline 
  $\nu$ &     0.009 &   0.457 &   3.0\\ \hline
  \end{tabular}
  \caption{Bias MSE Example $n=250$}
\end{table}
\subsection{Further Considerations}
In all cases, a Nelder-Mead method was selected for direct optimization in order to obtain the MLEs. There was not much investigation into other methods, but in a few trials, the Conjugate-Gradient method seemed to correct the numerical errors seen in the MLEs for $\lambda$. Further investigation should be done before concluding the Conjugate-Gradient method outperforms the Nelder-Mead method. When trying to find the MLE of a particular data set using the MVEW model in (\ref{mvewcdf}), it is suggested that different optimization procedures should be tried, with different starting values for each procedure, and then use a grid search to find the parameter values that optimize the likelihood function as defined in (\ref{likelihood}).
\section{Confidence Region Procedures}
In this section, each confidence region procedure was investigated with a simulation study. The performance of each procedure was assessed by observing estimated coverage probabilities.
\subsection{The Mahalanobis Distance Based on the Information Matrix}
In this simulation study, the procedure in (\ref{infoconf}) was applied to the model in (\ref{mvewcdf}) in which the test statistic in (\ref{mdist}) was compared to a $\chi^2_{0.05,(4)}$ critical value. The estimated coverage probabilities were observed for each set of parameters. It can be seen in table \ref{confreg50}, \ref{confreg100}, and \ref{confreg250} the estimated coverage probabilities were far below the $0.95$ coverage that would be expected. This would indicate the in the situations observed, the Mahalanobis distance is inappropriate as a method of obtaining confidence regions.\\
Another thing to note is that the observed information matrix was not always positive definite, and thus the Mahalanobis distance sometimes gave a negative value. In these cases, those values were not used in the estimated coverage probabilities. This is probably due to numerical error, since Berger (1985) noted that the information matrix is positive-definite if the the likelihood function is concave over a convex set. This phenomenon decreases in frequency as the sample size increases, since the likelihoods become less flat in these situations. \\
In all cases, an increase in sample size does seems to lead to the estimated coverage being closer to the nominal $0.95$. Even though this is the case, most of the situations observed do not seem to be close enough to the $0.95$ coverage to make this a viable method.
\subsection{The Mahalanobis Distance Based on the Bootstrap Estimate of the Covariance Matrix}
In this simulation study, the procedure in (\ref{bsconfreg}) was applied to the model in (\ref{mvewcdf}) in which the Mahalanobis distance in (\ref{bschisq}) was compared to a $\chi^2_{0.05,(4)}$ critical value. Estimated coverage probabilities were computed to assess the performance of this procedure to see if they were close to the nominal $0.95$ coverage.\\
In table \ref{confreg50}, \ref{confreg100}, and \ref{confreg250}, the procedure in (\ref{infoconf}) performed far worse than the procedure in (\ref{bsconfreg}) in the fact the coverage probabilities were far lower than the nominal $0.95$. The reason for this is the information matrix underestimated the standard errors of the individual parameters. This caused (\ref{infoconf}) to produce regions that were not large enough to capture a 0.95 coverage probability. In this case, the bootstrap estimated covariance matrix in (\ref{bscovariance}) tends to capture the covariance of the parameters of (\ref{mvewcdf}) and produce more liberal confidence regions.\\
One thing that is a very noticeable advantage to using the bootstrap covariance matrix is that the bootstrap covariance matrix will be guaranteed to be positive semidefinite, and it will almost always be of full rank and thus it will be positive definite. These will lead to results that are not out of the ordinary. 
\subsection{The Likelihood Ratio Test}
In the simulation study, a likelihood ratio test was performed using (\ref{LRconf}) on the model in (\ref{mvewcdf}), in which the Likelihood ratio in (\ref{LRteststat}) is compared to a $\chi^2_{0.05,(4)}$.  \\
In most cases (that can be seen in table \ref{confreg50}, \ref{confreg100}, and \ref{confreg250}), the likelihood ratio outperforms the bootstrap procedure, being on average closer to the expected $0.95$ coverage probability. The likelihood ratio test tends to produce more liberal confidence regions than the Mahalanobis distance based on the bootstrap. This can be attributed to the flat likelihood of many of the parameters, where large difference in the parameters values will still have relatively small changes to the values of the likelihood function.\\
In the situations in which the assumption of asymptotic normality is not valid $(\nu=1)$, the confidence region in (\ref{bsconfreg}) out performs the likelihood ratio test. In this situation, the procedure in (\ref{bsconfreg}) seems to be more robust against the assumption of asymptotic normality. Further study should be done to see if the procedure in (\ref{bsconfreg}) is more robust to the assumption of asymptotic normality than the procedure in (\ref{LRconf}) in general.
\subsection{Sample Size Considerations}
Even in the case of $n=250$, the procedure in (\ref{infoconf}) produced inadequate confidence regions. Although much larger sample sizes have not been studied, it is suggested to avoid this procedure for this particular model even if the sample size is very large, as the procedures in (\ref{bsconfreg}) and (\ref{LRconf}) produce far superior confidence regions.\\
The procedure in (\ref{bsconfreg}) typically produces adequate coverages at $n=100$. At $n=250$, the coverages are almost always close to the 0.95 coverage probability that is expected.\\
The procedure in (\ref{LRconf}) produced coverages very close to the expected $0.95$ coverage probability even when $n=50$, except in the case when $\nu=1$, where it typically gave confidence regions that were too conservative. When $n=250$, this problem becomes, for the most part, corrected.\\
\subsection{Some Tabular Example}
 Table 4.3 provides the estimated coverage probabilities for a small subset of the simulation study for each of the confidence regions discussed in Chapter 3. Complete tables can be found in Appendix A.2. The first column provides the estimated coverage probability generated from the Mahalanobis distance using the observed information matrix in (\ref{infoconf}). The second column provides the estimated coverage probability from the Mahalanobis distance using the bootstrap estimated covariance matrix in (\ref{bscovariance}), and then using the confidence region procedure found in (\ref{bsconfreg}). The third column provides the coverage probabilities for the likelihood ratio test found in (\ref{LRconf}). The fourth through seventh columns provide the values of the parameters from the model found in (\ref{mvewcdf}). In some cases, the Mahalanobis distance based on the information matrix was negative definite. If this happened, the sample was not used to compute the coverage probabilities. The eighth column provides the number of Mahalanobis distance based on the information matrix that were used to compute the coverage probability (with a maximum of 250).  
 \begin{table}[H]
 \centering
\begin{tabular}{|r|r|r|c|c|c|c|c|}
\hline
Information & Bootstrap & Ratio & $\lambda$ & $\alpha$ & $\theta$ & $\nu$ & $N$\\ \hline
\multicolumn{8}{|c|}{$n=50$}\\ \hline
 0.788 &   0.948 &   0.988 &   0.5 &   1.0 &   3.0 &   1.0 & 250\\ \hline
 0.628 &   0.908 &   0.944 &   1.0 &   0.5 &   1.0 &   2.0 & 250\\ \hline
 0.700 &   0.872 &   0.944 &   1.0 &   1.0 &   0.5 &   2.0 & 247\\ \hline
\multicolumn{8}{|c|}{$n=100$}\\ \hline
 0.836 &   0.960 &   0.976 &   0.5 &   1.0 &   3.0 &   1.0 & 250\\ \hline
 0.752 &   0.904 &   0.936 &   1.0 &   0.5 &   1.0 &   2.0 & 250\\ \hline
 0.788 &   0.924 &   0.964 &   1.0 &   1.0 &   0.5 &   2.0 & 250\\ \hline
\multicolumn{8}{|c|}{$n=250$}\\ \hline
 0.904 &   0.952 &   0.980 &   0.5 &   1.0 &   3.0 &   1.0 & 250\\ \hline
 0.836 &   0.924 &   0.968 &   1.0 &   0.5 &   1.0 &   2.0 & 250\\ \hline
 0.888 &   0.924 &   0.944 &   1.0 &   1.0 &   0.5 &   2.0 & 250\\ \hline
\end{tabular}
\caption{Confidence Region Example}
\end{table}
\subsection{Further Consideration}
In each situation, an estimated coverage probability was given in order to assess which procedure was most appropriate. In a real data setting, the practitioner should estimate the coverage probability for their particular situation by using bootstrap based methods to estimate the coverage for the method they are using. If the method they are using is inappropriate, they should use a different method.\\
In computing the test statistic in (\ref{bsasymptotic}), one should never compute the inverse of the covariance matrix. In order to find
\[
\hat{\Sigma}^{*-1}(\hat{\bm{\theta}}-\bm{\theta}),
\]
one should solve the equation
\[
\hat{\Sigma}^*\bm{x}=(\hat{\bm{\theta}}-\bm{\theta}),
\]
where $\bm{x}\in\mathbb{R}^d$, since the latter procedure in more numerically stable. \\
Another thing that was not discussed in this paper is a percentile bootstrap based on the bootstrap statistic 
\[
(\hat{\bm{\theta}}^*_b-\hat{\bm{\theta}})^\top\hat{\Sigma}^{*-1}(\hat{\bm{\theta}}^*_b-\hat{\bm{\theta}}),\,b=1,...B.
\]  
Such a statistic is worthy of investigation.
\section{Is There a Need for Bias Correction?} 
In this simulation study, 250 bootstrap $z$-intervals and  bootstrap $\text{BC}_a$ intervals were computed for each combination of values of parameters. For each collection of intervals, an average length and estimated coverage probability was given in order to compare the two intervals to see if there is a need for bias correction in the confidence region procedures. Each interval was computed so that the nominal coverage was $0.95$.\\
In the case when $\nu=1$, the $\text{BC}_a$ interval sometimes would not evaluate, due to the fact that $\nu=1$ is in the boundary of the parameter space. In these cases, those intervals were thrown out.
\subsection{Differences in Length and Coverage}
It can be seen in most cases in table \ref{bstab50}, \ref{bstab100}, and \ref{bstab250}, there was very similar coverage probabilities and average lengths. In the cases when $\nu=1$, the $z$-confidence interval seemed to be overly conservative, whereas the $\text{BC}_a$ interval tended to give more liberal coverage probabilities.\\
For small sample sizes, there tends to be larger differences in both the coverage probabilities, and the average lengths. Typically, the $z$-interval has a smaller length than the $\text{BC}_a$ interval, but overly conservative coverage probabilities. This is due to the fact that if the interval was outside the parameter space, it was truncated. In most cases, under small sample sizes, the $\text{BC}_a$ interval seemed to have coverages closer to the $0.95$ whereas the the $z$-interval tends to be overly conservative. For larger sample sizes $(n\geq100)$, the differences become negligible.\\
In the case of $n=50$, there is small need for bias correction, but this can typically be attributed to the fact that the asymptotics have not ``kicked in''. For larger $n$, the differences are negligible and there does not seem to be much of a need for it. See table 4.4 for an example. Complete tables are provided in appendix A.3.
\subsection{Some Tabular Examples}
Table 4.4 give the average lengths and estimated coverage probabilities for the bootstrap z-interval and the bootstrap $\text{BC}_a$ interval. Each collection of parameters are separated by a double horizontal slash. The first column provides the parameter in question. The second and third column provide the average length and estimated coverage probability of the bootstrap z-interval respectively. The fourth and fifth column provide the average length and estimated coverage of the bootstrap $\text{BC}_a$ interval respectively. In some cases, not all $\text{BC}_a$ intervals could be used. The fourth column provides the number of $\text{BC}_a$ intervals that were used to compute the the average length and to compute coverage probabilities (with a maximum of 250). The final column provides the value of the parameter in question that the sample
\begin{table}[H]
\centering
\begin{tabular}{|c|r|r|r|r|c|c|}
  \hline
    \multicolumn{7}{|c|}{$n=50$}\\ \hline
Par & ZCI Len. & ZCI Cov. & BCACI Len. & BCACI Cov. & BCACI Used & True \\ 
  \hline
  $\lambda$ & 35.182 & 0.940 & 30.303 & 0.924 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.820 & 0.916 & 0.760 & 0.924 & 250 & 0.5 \\ \hline
  $\theta$ & 1.022 & 0.920 & 1.102 & 0.928 & 250 & 0.5 \\ \hline
  $\nu$ & 0.188 & 0.996 & 0.312 & 0.524 & 138 & 1.0 \\ \hline\hline
  $\lambda$ & 5.992 & 0.968 & 8.620 & 0.940 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.418 & 0.952 & 1.324 & 0.940 & 250 & 1.0 \\ \hline
  $\theta$ & 26.983 & 0.972 & 34.994 & 0.936 & 250 & 3.0 \\ \hline
  $\nu$ & 1.830 & 0.912 & 1.886 & 0.928 & 250 & 3.0 \\ \hline\hline
  \multicolumn{7}{|c|}{$n=250$}\\ \hline
  $\lambda$ & 0.910 & 0.952 & 0.888 & 0.952 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.265 & 0.968 & 0.271 & 0.952 & 250 & 0.5 \\ \hline
  $\theta$ & 0.355 & 0.944 & 0.364 & 0.964 & 250 & 0.5 \\ \hline
  $\nu$ & 0.068 & 0.992 & 0.106 & 0.512 & 132 & 1.0 \\ \hline\hline
  $\lambda$ & 0.599 & 0.960 & 0.597 & 0.952 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.494 & 0.956 & 0.493 & 0.952 & 250 & 1.0 \\ \hline
  $\theta$ & 4.180 & 0.976 & 4.143 & 0.944 & 250 & 3.0 \\ \hline
  $\nu$ & 0.806 & 0.940 & 0.814 & 0.952 & 250 & 3.0 \\ \hline
  \end{tabular}
  \caption{Bootstrap $z$ vs. The Bootstrap $\text{BC}_a$ Example}
  \end{table}
  \subsection{Further Considerations}
  Although a need for bias correction has been seen for small samples in the confidence intervals, there has not been much work on bias correction for statistics like the one found in (\ref{bsasymptotic}). Percentile intervals can be created using the bootstrap statistic from \S 4.2.6, and the asymptotic distribution is known. From these facts, it would be of interest to create a method of bias correction using bootstrap statistics of that type.
\chapter{Conclusions and Summary}
Throughout this thesis, methods of constructing and performing inference on multivariate models has been discussed. Furthermore, assessment of these methods has been done with a simulation study. A summary of those conclusions will be provided in this chapter.
\section{Construction of a Multivariate Extension to a Univariate Distribution}
It is of interest to find a model for a multivariate distribution when the marginal distribution of the components is known and not necessarily Gaussian. In the case of the Gaussian distribution, the multivariate extension typically used is the $p$-dimensional normal distribution. In the case when the components do not have a Gaussian marginal distribution, the use of copulas is typically necessary.\\
In most cases, it is most practical to choose a copula model that can flexibly and adequately describe the dependence of the components. This thesis analyzed only one such copula model. There are of course many copula models that can be used in order to describe the dependence of a multivariate distribution.\\
Further research can be done by analyzing a class of copulas called the Archimedean copulas. It is of interest to determine the Archimedean copula associated with the exponentiated Weibull distribution and perform inference on that copula. The theory of Archimedean copulas can be found in Nelson (2006). 
\section{Simulation of a Multivariate Distribution}
When trying to simulate from a multivariate distribution, the Gibbs sampler is probably the most typical method of doing so. In the case of the $p$-dimensional normal distribution, the conditional distribution is known (and is in fact Gaussian), from which a Gibbs sampler can be achieved easily.\\
In the case when a $p$-dimension normal distribution is not used, other methods to perform a Gibbs sampler must be used.\\
This thesis has introduced two such methods. If the underlying Copula is known (or equivalently the joint cdf is known) the copula Gibbs sampler (algorithm 2) should be used. If, in the case where the underlying copula is not known, a Gibbs sampler can still be done using Monte Carlo methods (algorithm 3). \\
Further study could be done to improve the Monte Carlo Gibbs sampler. In the case presented in this thesis, a Monte Carlo integral was used to estimate the conditional cdf of a collection of random variables. There are other more sophisticated methods, such as quadrature, which require can estimate integrals of functions more precisely. Using quadrature, a smaller $\ell$ may be chosen to obtain the same accuracy, and thus will allow the algorithm to be more efficient.
\section{Inferential Procedures}
In parametric inference, maximum likelihood is typically used to estimate the parameters of a parametric model. In the case that was encountered with the model in (\ref{mvewcdf}), direct maximization was done. It was seen that when doing direct maximization using the Nelder-Mead method, there was substantial numeric error, especially when $\alpha=0.5$ in the model (\ref{mvewcdf}). It is suggested that further research be done on other numeric methods in finding MLEs, especially in the case of flat likelihood functions.
In more complicated cases, exact inference is not feasible, and thus asymptotic inference must be performed. In the three cases, the likelihood ratio test (\ref{LRteststat}) performed the best under the situations where asymptotic normality was applicable. In the case where asymptotic normality was not applicable, the Mahalanobis distance based on the bootstrap performed better. It is of interest to to investigate other methods of confidence regions, and further research should be done to see how those procedures might perform.
\section{Is There a Need for Bias Correction?}
For the case when the sample size is small, there were more significant differences in the estimated coverage probabilities of the $\text{BC}_a$ interval vs. the bootstrap $z$-interval. In the case when the sample sizes were larger, there seemed be a more negligible difference between the the two coverage probabilities. It is suggested to consider if there is a need for bias correction for only smaller sample sizes.
\section{Conclusions}
Many of the methods discussed in this paper were general methods that can be applied to many different constructions of multivariate extensions to univariate distributions. Of course, these methods should be studied for many different distributions with different copulas applied. If this is done, more general conclusions can be formed for future research.
\clearpage
\appendix
\chapter{Appendix: Tables}
\section{Bias and MSE table}
The following tables provides estimates of the bias (second column) and MSE (third column) for each parameter of the MVEW distribution in (\ref{mvewcdf}) for each collection of parameters. The parameter in question is provided (first column) and the value from which the distribution was simulated is provided (fourth column). Sample sizes for each set are provided as well (first row).
\clearpage
\begin{longtable}{|c|r|r|c|}
\hline
\multicolumn{4}{|c|}{n=50}\\ \hline
Parameter & Bias & MSE & True Value \\ \hline
 $\lambda$ &   0.364 &   1.596 &   0.5  \\ \hline
 $\alpha$ &   0.071 &   0.222 &   0.5  \\ \hline  
 $\theta$ &    0.019 &   0.248 &   0.5 \\ \hline 
  $\nu$ &      0.031 &   0.056 &   1.0 \\ \hline \hline
 $\lambda$ &   1.012 &   4.974 &  1.0 \\ \hline
 $\alpha$ &   0.022 &   0.158 &   0.5  \\ \hline
 $\theta$ &     0.059 &   0.264 &   0.5  \\ \hline
  $\nu$ &       0.030 &   0.051 &   1.0\\ \hline \hline
 $\lambda$ &   1.339 &   8.185 &   3.0  \\ \hline
 $\alpha$ &   0.054 &   0.212 &   0.5  \\ \hline 
 $\theta$ &     0.018 &   0.215 &   0.5  \\ \hline 
  $\nu$ &       0.031 &   0.059 &   1.0\\ \hline \hline
 $\lambda$ &   0.062 &   0.267 &   0.5  \\ \hline
 $\alpha$ &   0.101 &   0.417 &  1.0 \\ \hline 
 $\theta$ &     0.032 &   0.222 &   0.5  \\ \hline 
  $\nu$ &       0.028 &   0.048 &   1.0\\ \hline \hline
 $\lambda$ &   0.156 &   0.673 &  1.0 \\ \hline
 $\alpha$ &   0.104 &   0.433 &  1.0 \\ \hline 
 $\theta$ &     0.037 &   0.241 &   0.5  \\ \hline
  $\nu$ &       0.028 &   0.053 &   1.0\\ \hline \hline
 $\lambda$ &   0.347 &   2.291 &   3.0  \\ \hline
 $\alpha$ &   0.091 &   0.358 &  1.0 \\ \hline
 $\theta$ &     0.026 &   0.244 &   0.5  \\ \hline
  $\nu$ &       0.032 &   0.055 &   1.0\\ \hline \hline
 $\lambda$ &   0.007 &   0.071 &   0.5  \\ \hline
 $\alpha$ &  0.315 &   1.277 &   3.0  \\ \hline  
 $\theta$ &   0.028 &   0.230 &   0.5  \\ \hline  
  $\nu$ &    0.031 &   0.056 &   1.0\\ \hline \hline
 $\lambda$ &   0.032 &   0.182 &  1.0 \\ \hline
 $\alpha$ &   0.259 &   1.267 &   3.0  \\ \hline
 $\theta$ &    0.047 &   0.277 &   0.5  \\ \hline
  $\nu$ &     0.031 &   0.054 &   1.0\\ \hline \hline
 $\lambda$ &   0.023 &   0.418 &   3.0  \\ \hline
 $\alpha$ &  0.369 &   1.297 &   3.0  \\ \hline
 $\theta$ &   0.012 &   0.215 &   0.5  \\ \hline
  $\nu$ &     0.026 &   0.052 &   1.0\\ \hline \hline
 $\lambda$ &   0.330 &   1.268 &   0.5  \\ \hline
 $\alpha$ &  0.036 &   0.159 &   0.5  \\ \hline 
 $\theta$ &   0.090 &   0.514 &  1.0 \\ \hline
  $\nu$ &     0.031 &   0.054 &   1.0\\ \hline \hline
 $\lambda$ &   0.520 &   2.343 &  1.0 \\ \hline
 $\alpha$ &   0.028 &   0.165 &   0.5  \\ \hline
 $\theta$ &     0.094 &   0.515 &  1.0 \\ \hline
  $\nu$ &      0.027 &   0.051 &   1.0\\ \hline \hline
 $\lambda$ &   7.221 & 130.101 &   3.0  \\ \hline
 $\alpha$ &   0.017 &   0.128 &   0.5  \\ \hline
 $\theta$ &     0.133 &   0.729 &  1.0 \\ \hline
  $\nu$ &      0.021 &   0.040 &   1.0\\ \hline \hline
 $\lambda$ &   0.074 &   0.337 &   0.5  \\ \hline
 $\alpha$ &   0.081 &   0.352 &  1.0 \\ \hline
 $\theta$ &    0.097 &   0.546 &  1.0 \\ \hline
  $\nu$ &     0.028 &   0.054 &   1.0\\ \hline  \hline
 $\lambda$ &   0.273 &   1.385 &  1.0 \\ \hline
 $\alpha$ &   0.061 &   0.336 &  1.0 \\ \hline
 $\theta$ &    0.160 &   0.744 &  1.0 \\ \hline
  $\nu$ &    0.037 &   0.061 &   1.0\\ \hline \hline
 $\lambda$ &   0.391 &   3.521 &   3.0  \\ \hline
 $\alpha$ &   0.101 &   0.376 &  1.0 \\ \hline
 $\theta$ &     0.071 &   0.695 &  1.0 \\ \hline
  $\nu$ &      0.029 &   0.057 &   1.0\\ \hline \hline
 $\lambda$ &   0.023 &   0.122 &   0.5  \\ \hline
 $\alpha$ &  0.156 &   0.994 &   3.0  \\ \hline
 $\theta$ &   0.229 &   1.479 &  1.0 \\ \hline
  $\nu$ &   0.036 &   0.062 &   1.0\\ \hline \hline
 $\lambda$ &   0.007 &   0.141 &  1.0 \\ \hline
 $\alpha$ &   0.303 &   1.060 &   3.0  \\ \hline
 $\theta$ &     0.041 &   0.500 &  1.0 \\ \hline
  $\nu$ &      0.027 &   0.054 &   1.0\\ \hline \hline
 $\lambda$ &   0.090 &   0.470 &   3.0  \\ \hline
 $\alpha$ &   0.133 &   0.873 &   3.0  \\ \hline
 $\theta$ &     0.133 &   0.628 &  1.0 \\ \hline
  $\nu$ &      0.028 &   0.049 &   1.0\\ \hline \hline
 $\lambda$ &   0.735 &   4.349 &   0.5  \\ \hline
 $\alpha$ &  0.027 &   0.142 &   0.5  \\ \hline 
 $\theta$ &   0.627 &   3.293 &   3.0  \\ \hline
  $\nu$ &    0.030 &   0.054 &   1.0\\ \hline \hline
 $\lambda$ &   0.686 &   2.992 &  1.0 \\ \hline
 $\alpha$ &  0.024 &   0.144 &   0.5  \\ \hline
 $\theta$ &   0.533 &   2.394 &   3.0  \\ \hline  
  $\nu$ &   0.025 &   0.050 &   1.0\\ \hline \hline
 $\lambda$ &   1.463 &  10.350 &   3.0  \\ \hline
 $\alpha$ &  0.028 &   0.136 &   0.5  \\ \hline
 $\theta$ &   0.295 &   1.997 &   3.0  \\ \hline
  $\nu$ &    0.027 &   0.052 &   1.0\\ \hline \hline
 $\lambda$ &   0.137 &   0.516 &   0.5  \\ \hline
 $\alpha$ &  0.020 &   0.254 &  1.0 \\ \hline
 $\theta$ &   0.811 &   3.445 &   3.0  \\ \hline
  $\nu$ &    0.030 &   0.053 &   1.0\\ \hline \hline
 $\lambda$ &   0.187 &   0.786 &  1.0 \\ \hline
 $\alpha$ &   0.039 &   0.272 &  1.0 \\ \hline
 $\theta$ &     0.712 &   3.345 &   3.0  \\ \hline
  $\nu$ &      0.032 &   0.053 &   1.0\\ \hline \hline
 $\lambda$ &   0.594 &   3.294 &   3.0  \\ \hline
 $\alpha$ &   0.030 &   0.245 &  1.0 \\ \hline
 $\theta$ &     0.557 &   2.791 &   3.0  \\ \hline
  $\nu$ &      0.034 &   0.067 &   1.0\\ \hline \hline
 $\lambda$ &   0.021 &   0.107 &   0.5  \\ \hline
 $\alpha$ &   0.157 &   0.884 &   3.0  \\ \hline
 $\theta$ &     0.904 &   4.524 &   3.0  \\ \hline
  $\nu$ &      0.025 &   0.046 &   1.0\\ \hline \hline
 $\lambda$ &   0.038 &   0.215 &  1.0 \\ \hline
 $\alpha$ &   0.179 &   0.960 &   3.0  \\ \hline
 $\theta$ &     0.821 &   4.253 &   3.0  \\ \hline
  $\nu$ &     0.032 &   0.057 &   1.0\\ \hline \hline
 $\lambda$ &   0.147 &   0.692 &   3.0  \\ \hline
 $\alpha$ &  0.111 &   0.844 &   3.0  \\ \hline
 $\theta$ &   1.027 &   5.756 &   3.0  \\ \hline
  $\nu$ &    0.029 &   0.054 &   1.0\\ \hline  \hline
 $\lambda$ &   0.323 &   1.170 &   0.5  \\ \hline
 $\alpha$ &   0.055 &   0.199 &   0.5  \\ \hline
 $\theta$ &    0.036 &   0.229 &   0.5  \\ \hline
   $\nu$ &    0.043 &   0.314 &   2.0\\ \hline \hline
 $\lambda$ &   0.746 &   3.342 &  1.0 \\ \hline
 $\alpha$ &   0.077 &   0.212 &   0.5  \\ \hline
 $\theta$ &     0.015 &   0.255 &   0.5  \\ \hline
  $\nu$ &      0.009 &   0.280 &   2.0\\ \hline \hline
 $\lambda$ &   4.166 &  31.349 &   3.0  \\ \hline
 $\alpha$ &   0.065 &   0.234 &   0.5  \\ \hline
 $\theta$ &     0.060 &   0.309 &   0.5  \\ \hline
  $\nu$ &     $-$0.024 &   0.284 &   2.0\\ \hline \hline
 $\lambda$ &   0.098 &   0.349 &   0.5  \\ \hline
 $\alpha$ &   0.115 &   0.450 &  1.0 \\ \hline
 $\theta$ &     0.036 &   0.252 &   0.5  \\ \hline
  $\nu$ &      0.015 &   0.286 &   2.0\\ \hline \hline
 $\lambda$ &   0.141 &   0.882 &  1.0 \\ \hline
 $\alpha$ &   0.193 &   0.544 &  1.0 \\ \hline
 $\theta$ &     0.004 &   0.257 &   0.5  \\ \hline
  $\nu$ &      0.033 &   0.318 &   2.0\\ \hline \hline
 $\lambda$ &   0.498 &   2.864 &   3.0  \\ \hline
 $\alpha$ &   0.183 &   0.550 &  1.0 \\ \hline
 $\theta$ &    0.013 &   0.249 &   0.5  \\ \hline
  $\nu$ &     0.035 &   0.302 &   2.0\\ \hline \hline
 $\lambda$ &   0.002 &   0.073 &   0.5  \\ \hline
 $\alpha$ &   0.490 &   1.593 &   3.0  \\ \hline
 $\theta$ &    0.007 &   0.234 &   0.5  \\ \hline
  $\nu$ &    0.040 &   0.287 &   2.0\\ \hline  \hline
 $\lambda$ &   0.032 &   0.210 &  1.0 \\ \hline
 $\alpha$ &   0.466 &   1.577 &   3.0  \\ \hline
 $\theta$ &     0.034 &   0.301 &   0.5  \\ \hline
  $\nu$ &     $-$0.026 &   0.279 &   2.0\\ \hline \hline
 $\lambda$ &   0.071 &   0.509 &   3.0  \\ \hline
 $\alpha$ &   0.389 &   1.335 &   3.0  \\ \hline
 $\theta$ &     0.034 &   0.259 &   0.5  \\ \hline
  $\nu$ &      0.030 &   0.297 &   2.0\\ \hline \hline
 $\lambda$ &   0.562 &   2.547 &   0.5  \\ \hline
 $\alpha$ &   0.050 &   0.180 &   0.5  \\ \hline
 $\theta$ &    0.163 &   0.743 &  1.0 \\ \hline
  $\nu$ &    $-$0.002 &   0.281 &   2.0\\ \hline \hline
 $\lambda$ &   1.514 &  13.480 &  1.0 \\ \hline
 $\alpha$ &   0.063 &   0.204 &   0.5  \\ \hline
 $\theta$ &     0.131 &   0.779 &  1.0 \\ \hline
  $\nu$ &      0.002 &   0.288 &   2.0\\ \hline \hline
 $\lambda$ &   2.196 &  12.485 &   3.0  \\ \hline
 $\alpha$ &   0.053 &   0.215 &   0.5  \\ \hline
 $\theta$ &     0.120 &   0.583 &  1.0 \\ \hline
  $\nu$ &     $-$0.020 &   0.281 &   2.0\\ \hline \hline
 $\lambda$ &   0.136 &   0.524 &   0.5  \\ \hline
 $\alpha$ &  0.120 &   0.572 &  1.0 \\ \hline
 $\theta$ &    0.170 &   0.750 &  1.0 \\ \hline
  $\nu$ &    $-$0.002 &   0.278 &   2.0\\ \hline \hline
 $\lambda$ &   0.290 &   1.341 &  1.0 \\ \hline
 $\alpha$ &   0.088 &   0.381 &  1.0 \\ \hline
 $\theta$ &     0.170 &   0.914 &  1.0 \\ \hline
  $\nu$ &     $-$0.012 &   0.284 &   2.0\\ \hline \hline
 $\lambda$ &   1.036 &   5.363 &   3.0  \\ \hline
 $\alpha$ &   0.096 &   0.403 &  1.0 \\ \hline
 $\theta$ &     0.197 &   0.913 &  1.0 \\ \hline
  $\nu$ &     $-$0.012 &   0.305 &   2.0\\ \hline \hline
 $\lambda$ &   0.019 &   0.088 &   0.5  \\ \hline
 $\alpha$ &   0.230 &   1.110 &   3.0  \\ \hline
 $\theta$ &     0.160 &   0.680 &  1.0 \\ \hline
  $\nu$ &      0.000 &   0.306 &   2.0\\ \hline \hline
 $\lambda$ &   0.031 &   0.200 &  1.0 \\ \hline
 $\alpha$ &   0.359 &   1.468 &   3.0  \\ \hline
 $\theta$ &     0.131 &   0.692 &  1.0 \\ \hline
  $\nu$ &      0.004 &   0.268 &   2.0\\ \hline \hline
 $\lambda$ &   0.098 &   0.536 &   3.0  \\ \hline
 $\alpha$ &   0.308 &   1.413 &   3.0  \\ \hline
 $\theta$ &     0.146 &   0.687 &  1.0 \\ \hline
  $\nu$ &     $-$0.001 &   0.278 &   2.0\\ \hline \hline
 $\lambda$ &   1.449 &   7.298 &   0.5  \\ \hline
 $\alpha$ &   0.032 &   0.196 &   0.5  \\ \hline
 $\theta$ &     1.495 &   6.360 &   3.0  \\ \hline
  $\nu$ &     $-$0.027 &   0.297 &   2.0\\ \hline \hline
 $\lambda$ &   1.772 &  10.037 &  1.0 \\ \hline
 $\alpha$ &   0.030 &   0.172 &   0.5  \\ \hline
 $\theta$ &     1.087 &   5.013 &   3.0  \\ \hline
  $\nu$ &      0.016 &   0.291 &   2.0\\ \hline \hline
 $\lambda$ &   6.331 &  60.299 &   3.0  \\ \hline
 $\alpha$ &   0.025 &   0.153 &   0.5  \\ \hline
 $\theta$ &     1.243 &   5.226 &   3.0  \\ \hline
  $\nu$ &     $-$0.025 &   0.281 &   2.0\\ \hline \hline
 $\lambda$ &   0.135 &   0.481 &   0.5  \\ \hline
 $\alpha$ &   0.054 &   0.306 &  1.0 \\ \hline
 $\theta$ &     0.931 &   3.889 &   3.0  \\ \hline
  $\nu$ &      0.017 &   0.287 &   2.0\\ \hline \hline
 $\lambda$ &   0.330 &   1.712 &  1.0 \\ \hline 
 $\alpha$ &   0.110 &   0.390 &  1.0 \\ \hline
 $\theta$ &     0.934 &   4.989 &   3.0  \\ \hline
  $\nu$ &      0.010 &   0.302 &   2.0\\ \hline \hline
 $\lambda$ &   1.408 &   7.619 &   3.0  \\ \hline
 $\alpha$ &   0.062 &   0.315 &  1.0 \\ \hline
 $\theta$ &     1.439 &   7.494 &   3.0  \\ \hline
  $\nu$ &     $-$0.006 &   0.298 &   2.0\\ \hline \hline
 $\lambda$ &   0.020 &   0.120 &   0.5  \\ \hline
 $\alpha$ &   0.362 &   1.263 &   3.0  \\ \hline
 $\theta$ &     1.208 &   6.775 &   3.0  \\ \hline
  $\nu$ &     $-$0.011 &   0.293 &   2.0\\ \hline \hline
 $\lambda$ &   0.097 &   0.479 &  1.0 \\ \hline
 $\alpha$ &   0.173 &   0.999 &   3.0  \\ \hline
 $\theta$ &     2.710 &  21.871 &   3.0  \\ \hline
  $\nu$ &      0.034 &   0.314 &   2.0\\ \hline \hline
 $\lambda$ &   0.207 &   1.067 &   3.0  \\ \hline
 $\alpha$ &   0.212 &   1.054 &   3.0  \\ \hline
 $\theta$ &     2.119 &  14.656 &   3.0  \\ \hline
  $\nu$ &      0.006 &   0.263 &   2.0\\ \hline \hline
 $\lambda$ &   0.717 &   3.033 &   0.5  \\ \hline
 $\alpha$ &   0.056 &   0.231 &   0.5  \\ \hline
 $\theta$ &     0.063 &   0.304 &   0.5  \\ \hline
  $\nu$ &     0.025 &   0.459 &   3.0\\ \hline  \hline
 $\lambda$ &   1.153 &   9.343 &  1.0 \\ \hline
 $\alpha$ &   0.075 &   0.228 &   0.5  \\ \hline
 $\theta$ &     0.029 &   0.285 &   0.5  \\ \hline
  $\nu$ &     $-$0.015 &   0.469 &   3.0\\ \hline \hline
 $\lambda$ &   4.173 &  38.875 &   3.0  \\ \hline
 $\alpha$ &   0.088 &   0.260 &   0.5  \\ \hline
 $\theta$ &     0.041 &   0.298 &   0.5  \\ \hline
  $\nu$ &      0.047 &   0.534 &   3.0\\ \hline \hline
 $\lambda$ &   0.131 &   0.552 &   0.5  \\ \hline
 $\alpha$ &   0.147 &   0.471 &  1.0 \\ \hline
 $\theta$ &     0.042 &   0.281 &   0.5  \\ \hline
  $\nu$ &     $-$0.042 &   0.501 &   3.0\\ \hline \hline
 $\lambda$ &   0.173 &   0.752 &  1.0 \\ \hline
 $\alpha$ &   0.161 &   0.476 &  1.0 \\ \hline
 $\theta$ &    0.022 &   0.253 &   0.5  \\ \hline
  $\nu$ &    $-$0.001 &   0.409 &   3.0\\ \hline \hline
 $\lambda$ &   0.965 &   4.621 &   3.0  \\ \hline
 $\alpha$ &  0.102 &   0.435 &  1.0 \\ \hline
 $\theta$ &   0.077 &   0.332 &   0.5  \\ \hline
  $\nu$ &    0.014 &   0.473 &   3.0\\ \hline \hline
 $\lambda$ &   0.023 &   0.117 &   0.5  \\ \hline
 $\alpha$ &   0.404 &   1.582 &   3.0  \\ \hline
 $\theta$ &     0.059 &   0.340 &   0.5  \\ \hline
  $\nu$ &     $-$0.028 &   0.433 &   3.0\\ \hline \hline
 $\lambda$ &   0.036 &   0.178 &  1.0 \\ \hline
 $\alpha$ &   0.387 &   1.340 &   3.0  \\ \hline
 $\theta$ &    0.042 &   0.273 &   0.5  \\ \hline
  $\nu$ &    0.008 &   0.443 &   3.0\\ \hline  \hline
 $\lambda$ &   0.170 &   0.662 &   3.0  \\ \hline
 $\alpha$ &   0.369 &   1.612 &   3.0  \\ \hline  
 $\theta$ &    0.073 &   0.346 &   0.5  \\ \hline
  $\nu$ &     0.025 &   0.467 &   3.0\\ \hline \hline
 $\lambda$ &   0.437 &   1.655 &   0.5  \\ \hline
 $\alpha$ &   0.039 &   0.167 &   0.5  \\ \hline 
 $\theta$ &     0.130 &   0.661 &  1.0 \\ \hline
  $\nu$ &      0.065 &   0.494 &   3.0\\ \hline \hline
 $\lambda$ &   1.119 &   4.874 &  1.0 \\ \hline
 $\alpha$ &  0.042 &   0.186 &   0.5  \\ \hline
 $\theta$ &    0.159 &   0.688 &  1.0 \\ \hline
  $\nu$ &    $-$0.012 &   0.518 &   3.0\\ \hline \hline
 $\lambda$ &   4.402 &  42.674 &   3.0  \\ \hline 
 $\alpha$ &   0.038 &   0.188 &   0.5  \\ \hline
 $\theta$ &     0.188 &   0.712 &  1.0 \\ \hline
  $\nu$ &      0.010 &   0.463 &   3.0\\ \hline \hline
 $\lambda$ &   0.228 &   1.426 &   0.5  \\ \hline
 $\alpha$ &   0.086 &   0.383 &  1.0 \\ \hline
 $\theta$ &    0.208 &   1.101 &  1.0 \\ \hline
  $\nu$ &    0.002 &   0.444 &   3.0\\ \hline \hline
 $\lambda$ &   0.126 &   0.644 &  1.0 \\ \hline
 $\alpha$ &   0.114 &   0.356 &  1.0 \\ \hline
 $\theta$ &     0.082 &   0.579 &  1.0 \\ \hline
  $\nu$ &     0.031 &   0.488 &   3.0\\ \hline \hline
 $\lambda$ &   0.531 &   3.318 &   3.0  \\ \hline
 $\alpha$ &  0.097 &   0.338 &  1.0\\ \hline  
 $\theta$ &   0.093 &   0.704 &  1.0\\ \hline 
  $\nu$ &    0.059 &   0.493 &   3.0\\ \hline \hline
 $\lambda$ &   0.017 &   0.096 &   0.5 \\ \hline
 $\alpha$ &   0.365 &   1.532 &   3.0 \\ \hline 
 $\theta$ &     0.140 &   0.746 &  1.0 \\ \hline 
  $\nu$ &     $-$0.009 &   0.454 &   3.0\\ \hline \hline
 $\lambda$ &   0.045 &   0.205 &  1.0\\ \hline
 $\alpha$ &  0.362 &   1.563 &   3.0  \\ \hline  
 $\theta$ &   0.184 &   0.792 &  1.0 \\ \hline
  $\nu$ &   $-$0.044 &   0.448 &   3.0\\ \hline \hline
 $\lambda$ &   0.150 &   0.703 &   3.0  \\ \hline
 $\alpha$ &   0.394 &   1.890 &   3.0  \\ \hline
 $\theta$ &     0.207 &   0.920 &  1.0 \\ \hline
  $\nu$ &     $-$0.007 &   0.453 &   3.0\\ \hline \hline
 $\lambda$ &   0.823 &   3.521 &   0.5  \\ \hline
 $\alpha$ &   0.026 &   0.146 &   0.5  \\ \hline
 $\theta$ &    1.019 &   4.364 &   3.0  \\ \hline
  $\nu$ &    $-$0.055 &   0.444 &   3.0\\ \hline \hline
 $\lambda$ &   3.079 &  20.046 &  1.0 \\ \hline
 $\alpha$ &  0.024 &   0.175 &   0.5  \\ \hline  
 $\theta$ &   1.655 &   6.843 &   3.0  \\ \hline 
  $\nu$ &   $-$0.054 &   0.485 &   3.0\\ \hline \hline
 $\lambda$ &   5.123 &  44.064 &   3.0  \\ \hline
 $\alpha$ &   0.039 &   0.171 &   0.5  \\ \hline
 $\theta$ &    1.047 &   4.432 &   3.0  \\ \hline 
  $\nu$ &     0.009 &   0.457 &   3.0\\ \hline \hline
 $\lambda$ &   0.248 &   0.980 &   0.5  \\ \hline
 $\alpha$ &  0.064 &   0.315 &  1.0 \\ \hline  
 $\theta$ &   1.432 &   6.889 &   3.0  \\ \hline 
  $\nu$ &   $-$0.002 &   0.490 &   3.0\\ \hline \hline
 $\lambda$ &   0.440 &   2.096 &  1.0 \\ \hline
 $\alpha$ &  0.059 &   0.312 &  1.0 \\ \hline  
 $\theta$ &   1.586 &   8.572 &   3.0  \\ \hline 
  $\nu$ &   $-$0.033 &   0.476 &   3.0\\ \hline \hline
 $\lambda$ &   0.855 &   4.088 &   3.0  \\ \hline
 $\alpha$ &   0.064 &   0.310 &  1.0 \\ \hline
 $\theta$ &     0.935 &   4.329 &   3.0  \\ \hline 
  $\nu$ &     $-$0.018 &   0.476 &   3.0\\ \hline \hline
 $\lambda$ &   0.027 &   0.131 &   0.5  \\ \hline
 $\alpha$ &   0.168 &   0.915 &   3.0  \\ \hline  
 $\theta$ &    1.293 &   7.138 &   3.0  \\ \hline 
  $\nu$ &     0.022 &   0.457 &   3.0\\ \hline \hline
 $\lambda$ &   0.078 &   0.301 &  1.0 \\ \hline
 $\alpha$ &   0.122 &   0.965 &   3.0  \\ \hline 
 $\theta$ &     1.858 &  10.987 &   3.0  \\ \hline 
  $\nu$ &      0.007 &   0.440 &   3.0\\ \hline \hline
 $\lambda$ &   0.136 &   0.698 &   3.0  \\ \hline
 $\alpha$ &   0.193 &   0.971 &   3.0  \\ \hline  
 $\theta$ &    1.094 &   5.197 &   3.0  \\ \hline  
  $\nu$ &    0.069 &   0.491 &   3.0\\ \hline
 \caption{Bias and MSE Table for $n=50$}
 \label{biasmse50}
  \end{longtable}

\clearpage
\begin{longtable}{|c|r|r|c|}
\hline
\multicolumn{4}{|c|}{n=100}\\ \hline
Parameter & Bias & MSE & True Value \\ \hline   
$\lambda$ &   0.133 &   0.491 &   0.5 \\ \hline
$\alpha$ &   0.022 &   0.117 &   0.5 \\ \hline
$\theta$ &     0.021 &   0.156 &   0.5 \\ \hline
 $\nu$ &       0.022 &   0.037 &   1.0 \\ \hline \hline
$\lambda$ &   0.196 &   1.322 &   1.0 \\ \hline
$\alpha$ &   0.033 &   0.115 &   0.5 \\ \hline
$\theta$ &     0.002 &   0.151 &   0.5 \\ \hline
 $\nu$ &       0.019 &   0.035 &   1.0 \\ \hline\hline
$\lambda$ &   0.319 &   1.748 &   3.0 \\ \hline 
$\alpha$ &  0.024 &   0.105 &   0.5 \\ \hline
$\theta$ &    0.002 &   0.114 &   0.5 \\ \hline
 $\nu$ &      0.019 &   0.032 &   1.0 \\ \hline\hline
$\lambda$ &   0.037 &   0.183 &   0.5 \\ \hline
$\alpha$ &   0.054 &   0.256 &   1.0 \\ \hline
$\theta$ &     0.018 &   0.162 &   0.5 \\ \hline
 $\nu$ &       0.024 &   0.040 &   1.0 \\ \hline\hline
$\lambda$ &   0.049 &   0.410 &   1.0 \\ \hline
$\alpha$ &   0.066 &   0.262 &   1.0 \\ \hline
$\theta$ &     0.008 &   0.173 &   0.5 \\ \hline
 $\nu$ &       0.022 &   0.039 &   1.0 \\ \hline\hline
$\lambda$ &   0.099 &   0.886 &   3.0 \\ \hline
$\alpha$ &   0.060 &   0.231 &   1.0 \\ \hline
$\theta$ &     0.005 &   0.138 &   0.5 \\ \hline
 $\nu$ &      0.024 &   0.037 &   1.0 \\ \hline\hline
$\lambda$ &   0.000 &   0.044 &   0.5 \\ \hline
$\alpha$ &   0.181 &   0.803 &   3.0 \\ \hline
$\theta$ &     0.001 &   0.131 &   0.5 \\ \hline
 $\nu$ &       0.018 &   0.034 &   1.0 \\ \hline \hline
$\lambda$ &   0.009 &   0.093 &   1.0 \\ \hline
$\alpha$ &   0.121 &   0.689 &   3.0 \\ \hline
$\theta$ &     0.016 &   0.146 &   0.5 \\ \hline
 $\nu$ &       0.018 &   0.033 &   1.0 \\ \hline\hline
$\lambda$ &   0.056 &   0.297 &   3.0 \\ \hline
$\alpha$ &   0.083 &   0.681 &   3.0 \\ \hline
$\theta$ &    0.029 &   0.152 &   0.5 \\ \hline
 $\nu$ &       0.021 &   0.036 &   1.0 \\ \hline\hline
$\lambda$ &   0.119 &   0.504 &   0.5 \\ \hline
$\alpha$ &   0.017 &   0.088 &   0.5 \\ \hline
$\theta$ &     0.035 &   0.330 &   1.0 \\ \hline
 $\nu$ &       0.022 &   0.036 &   1.0 \\ \hline\hline
$\lambda$ &   0.209 &   1.010 &   1.0 \\ \hline
$\alpha$ &   0.020 &   0.096 &   0.5 \\ \hline
$\theta$ &     0.041 &   0.343 &   1.0 \\ \hline
 $\nu$ &       0.018 &   0.032 &   1.0 \\ \hline\hline
$\lambda$ &   0.662 &   3.866 &   3.0 \\ \hline
$\alpha$ &   0.014 &   0.089 &   0.5 \\ \hline 
$\theta$ &    0.041 &   0.320 &   1.0 \\ \hline
  $\nu$ &     0.022 &   0.036 &   1.0 \\ \hline\hline
$\lambda$ &   0.056 &   0.210 &   0.5 \\ \hline 
$\alpha$ &  0.023 &   0.213 &   1.0 \\ \hline 
$\theta$ &   0.082 &   0.394 &   1.0 \\ \hline
 $\nu$ &     0.020 &   0.036 &   1.0 \\ \hline\hline
$\lambda$ &   0.088 &   0.374 &   1.0 \\ \hline
$\alpha$ &   0.019 &   0.192 &   1.0 \\ \hline
$\theta$ &     0.072 &   0.368 &   1.0 \\ \hline
 $\nu$ &       0.022 &   0.039 &   1.0 \\ \hline \hline
$\lambda$ &   0.194 &   0.928 &   3.0 \\ \hline
$\alpha$ &   0.017 &   0.183 &   1.0 \\ \hline
$\theta$ &     0.051 &   0.303 &   1.0 \\ \hline 
  $\nu$ &     0.023 &   0.035 &   1.0 \\ \hline\hline
$\lambda$ &   0.004 &   0.044 &   0.5 \\ \hline
$\alpha$ &   0.096 &   0.533 &   3.0 \\ \hline
$\theta$ &     0.026 &   0.303 &   1.0 \\ \hline
  $\nu$ &      0.017 &   0.032 &   1.0 \\ \hline\hline
$\lambda$ &   0.015 &   0.107 &   1.0 \\ \hline
$\alpha$ &  0.060 &   0.580 &   3.0 \\ \hline
$\theta$ &     0.056 &   0.374 &   1.0 \\ \hline
  $\nu$ &      0.023 &   0.037 &   1.0 \\ \hline\hline
$\lambda$ &   0.037 &   0.293 &   3.0 \\ \hline
$\alpha$ &   0.116 &   0.637 &   3.0 \\ \hline 
$\theta$ &    0.049 &   0.340 &   1.0 \\ \hline
  $\nu$ &     0.019 &   0.036 &   1.0 \\ \hline\hline
$\lambda$ &   0.138 &   0.638 &   0.5 \\ \hline
$\alpha$ &   0.009 &   0.081 &   0.5 \\ \hline
$\theta$ &     0.258 &   1.335 &   3.0 \\ \hline
 $\nu$ &       0.017 &   0.034 &   1.0 \\ \hline\hline
$\lambda$ &   0.232 &   1.028 &   1.0 \\ \hline
$\alpha$ &   0.009 &   0.080 &   0.5 \\ \hline
$\theta$ &     0.209 &   1.252 &   3.0 \\ \hline
 $\nu$ &       0.018 &   0.034 &   1.0 \\ \hline\hline
$\lambda$ &   1.389 &   8.534 &   3.0 \\ \hline
$\alpha$ &   0.006 &   0.080 &   0.5 \\ \hline
$\theta$ &     0.316 &   1.606 &   3.0 \\ \hline
 $\nu$ &       0.019 &   0.034 &   1.0 \\ \hline \hline
$\lambda$ &   0.024 &   0.165 &   0.5 \\ \hline
$\alpha$ &   0.034 &   0.172 &   1.0 \\ \hline
$\theta$ &     0.184 &   1.277 &   3.0 \\ \hline
 $\nu$ &       0.018 &   0.032 &   1.0 \\ \hline\hline
$\lambda$ &   0.086 &   0.516 &   1.0 \\ \hline
$\alpha$ &   0.025 &   0.173 &   1.0 \\ \hline
$\theta$ &     0.269 &   1.754 &   3.0 \\ \hline
 $\nu$ &       0.019 &   0.036 &   1.0 \\ \hline\hline
$\lambda$ &   0.322 &   1.731 &   3.0 \\ \hline
$\alpha$ &   0.017 &   0.177 &   1.0 \\ \hline
$\theta$ &     0.377 &   1.929 &   3.0 \\ \hline
 $\nu$ &       0.019 &   0.034 &   1.0 \\ \hline\hline
$\lambda$ &   0.007 &   0.058 &   0.5 \\ \hline
$\alpha$ &   0.069 &   0.520 &   3.0 \\ \hline
$\theta$ &     0.287 &   1.591 &   3.0 \\ \hline
 $\nu$ &       0.017 &   0.030 &   1.0 \\ \hline\hline
$\lambda$ &   0.018 &   0.118 &   1.0 \\ \hline
$\alpha$ &   0.057 &   0.531 &   3.0 \\ \hline
$\theta$ &     0.384 &   2.128 &   3.0 \\ \hline 
 $\nu$ &      0.024 &   0.040 &   1.0 \\ \hline\hline
$\lambda$ &   0.048 &   0.370 &   3.0 \\ \hline
$\alpha$ &   0.061 &   0.554 &   3.0 \\ \hline
$\theta$ &     0.322 &   1.783 &   3.0 \\ \hline 
 $\nu$ &      0.023 &   0.037 &   1.0 \\ \hline \hline
$\lambda$ &   0.149 &   0.543 &   0.5 \\ \hline 
$\alpha$ &  0.032 &   0.123 &   0.5 \\ \hline 
$\theta$ &   0.011 &   0.161 &   0.5 \\ \hline
  $\nu$ &   $-$0.003 &   0.206 &   2.0 \\ \hline\hline
$\lambda$ &   0.345 &   1.410 &   1.0 \\ \hline 
$\alpha$ &  0.030 &   0.130 &   0.5 \\ \hline
$\theta$ &    0.018 &   0.171 &   0.5 \\ \hline
  $\nu$ &     0.009 &   0.198 &   2.0 \\ \hline\hline
$\lambda$ &   1.062 &   5.147 &   3.0 \\ \hline
$\alpha$ &   0.035 &   0.139 &   0.5 \\ \hline
$\theta$ &     0.021 &   0.183 &   0.5 \\ \hline
  $\nu$ &     $-$0.021 &   0.195 &   2.0 \\ \hline\hline
$\lambda$ &   0.043 &   0.281 &   0.5 \\ \hline
$\alpha$ &   0.076 &   0.291 &   1.0 \\ \hline
$\theta$ &     0.016 &   0.204 &   0.5 \\ \hline
  $\nu$ &      0.011 &   0.200 &   2.0 \\ \hline\hline
$\lambda$ &   0.065 &   0.368 &   1.0 \\ \hline
$\alpha$ &   0.059 &   0.252 &   1.0 \\ \hline
$\theta$ &     0.003 &   0.155 &   0.5 \\ \hline
  $\nu$ &     $-$0.005 &   0.212 &   2.0 \\ \hline\hline
$\lambda$ &   0.150 &   1.121 &   3.0 \\ \hline
$\alpha$ &   0.083 &   0.287 &   1.0 \\ \hline 
$\theta$ &    0.002 &   0.158 &   0.5 \\ \hline 
 $\nu$ &     0.015 &   0.192 &   2.0 \\ \hline\hline
$\lambda$ &   0.006 &   0.055 &   0.5 \\ \hline
$\alpha$ &   0.169 &   0.673 &   3.0 \\ \hline
$\theta$ &     0.013 &   0.165 &   0.5 \\ \hline
 $\nu$ &      $-$0.007 &   0.211 &   2.0 \\ \hline \hline
$\lambda$ &   0.004 &   0.108 &   1.0 \\ \hline 
$\alpha$ &  0.199 &   0.755 &   3.0 \\ \hline 
$\theta$ &   0.009 &   0.158 &   0.5 \\ \hline 
 $\nu$ &    0.014 &   0.195 &   2.0 \\ \hline\hline
$\lambda$ &   0.068 &   0.371 &   3.0 \\ \hline
$\alpha$ &   0.160 &   0.780 &   3.0 \\ \hline
$\theta$ &     0.026 &   0.187 &   0.5 \\ \hline
 $\nu$ &      $-$0.001 &   0.209 &   2.0 \\ \hline\hline
$\lambda$ &   0.134 &   0.540 &   0.5 \\ \hline
$\alpha$ &   0.019 &   0.101 &   0.5 \\ \hline
$\theta$ &     0.042 &   0.366 &   1.0 \\ \hline 
  $\nu$ &     0.016 &   0.205 &   2.0 \\ \hline\hline
$\lambda$ &   0.241 &   1.093 &   1.0 \\ \hline
$\alpha$ &   0.029 &   0.107 &   0.5 \\ \hline
$\theta$ &     0.016 &   0.334 &   1.0 \\ \hline
 $\nu$ &       0.009 &   0.214 &   2.0 \\ \hline\hline
$\lambda$ &   0.879 &   3.705 &   3.0 \\ \hline
$\alpha$ &   0.022 &   0.110 &   0.5 \\ \hline
$\theta$ &     0.054 &   0.372 &   1.0 \\ \hline
 $\nu$ &       0.024 &   0.235 &   2.0 \\ \hline\hline
$\lambda$ &   0.047 &   0.181 &   0.5 \\ \hline
$\alpha$ &   0.027 &   0.202 &   1.0 \\ \hline
$\theta$ &     0.074 &   0.356 &   1.0 \\ \hline
 $\nu$ &      $-$0.003 &   0.184 &   2.0 \\ \hline\hline
$\lambda$ &   0.119 &   0.417 &   1.0 \\ \hline
$\alpha$ &   0.027 &   0.219 &   1.0 \\ \hline
$\theta$ &     0.092 &   0.406 &   1.0 \\ \hline
 $\nu$ &      $-$0.011 &   0.198 &   2.0 \\ \hline \hline
$\lambda$ &   0.103 &   1.022 &   3.0 \\ \hline
$\alpha$ &   0.063 &   0.234 &   1.0 \\ \hline
$\theta$ &     0.024 &   0.349 &   1.0 \\ \hline
 $\nu$ &       0.003 &   0.199 &   2.0 \\ \hline\hline
$\lambda$ &   0.001 &   0.050 &   0.5 \\ \hline
$\alpha$ &   0.161 &   0.640 &   3.0 \\ \hline
$\theta$ &     0.009 &   0.310 &   1.0 \\ \hline
 $\nu$ &       0.015 &   0.215 &   2.0 \\ \hline\hline
$\lambda$ &   0.027 &   0.117 &   1.0 \\ \hline
$\alpha$ &   0.064 &   0.646 &   3.0 \\ \hline
$\theta$ &     0.095 &   0.383 &   1.0 \\ \hline
 $\nu$ &      $-$0.007 &   0.205 &   2.0 \\ \hline\hline
$\lambda$ &   0.077 &   0.337 &   3.0 \\ \hline
$\alpha$ &   0.096 &   0.669 &   3.0 \\ \hline
$\theta$ &     0.078 &   0.399 &   1.0 \\ \hline
 $\nu$ &      $-$0.004 &   0.209 &   2.0 \\ \hline\hline
$\lambda$ &   0.500 &  2.481 &   0.5 \\ \hline
$\alpha$ &   0.002 &   0.109 &   0.5 \\ \hline
$\theta$ &     0.745 &   2.811 &   3.0 \\ \hline
 $\nu$ &       0.014 &   0.192 &   2.0 \\ \hline\hline
$\lambda$ &   0.412 &   1.869 &   1.0 \\ \hline
$\alpha$ &   0.031 &   0.114 &   0.5 \\ \hline
$\theta$ &     0.256 &   1.794 &   3.0 \\ \hline
 $\nu$ &      $-$0.015 &   0.198 &   2.0 \\ \hline\hline
$\lambda$ &   1.738 &   7.921 &   3.0 \\ \hline
$\alpha$ &   0.011 &   0.107 &   0.5 \\ \hline
$\theta$ &     0.541 &   2.049 &   3.0 \\ \hline
 $\nu$ &      $-$0.009 &   0.194 &   2.0 \\ \hline \hline
$\lambda$ &   0.040 &   0.216 &   0.5 \\ \hline
$\alpha$ &   0.046 &   0.216 &   1.0 \\ \hline
$\theta$ &     0.264 &   1.578 &   3.0 \\ \hline
 $\nu$ &       0.001 &   0.193 &   2.0 \\ \hline\hline
$\lambda$ &   0.112 &   0.554 &   1.0 \\ \hline
$\alpha$ &   0.049 &   0.238 &   1.0 \\ \hline 
$\theta$ &    0.342 &   2.029 &   3.0 \\ \hline
 $\nu$ &     $-$0.007 &   0.186 &   2.0 \\ \hline\hline
$\lambda$ &   0.318 &   1.420 &   3.0 \\ \hline
$\alpha$ &   0.029 &   0.209 &   1.0 \\ \hline
$\theta$ &     0.373 &   1.680 &   3.0 \\ \hline
 $\nu$ &       0.006 &   0.204 &   2.0 \\ \hline\hline
$\lambda$ &   0.007 &   0.067 &   0.5 \\ \hline
$\alpha$ &   0.135 &   0.672 &   3.0 \\ \hline
$\theta$ &     0.362 &   1.844 &   3.0 \\ \hline
 $\nu$ &       0.015 &   0.197 &   2.0 \\ \hline\hline
$\lambda$ &   0.035 &   0.138 &   1.0 \\ \hline
$\alpha$ &   0.027 &   0.564 &   3.0 \\ \hline
$\theta$ &     0.580 &   2.230 &   3.0 \\ \hline
 $\nu$ &      $-$0.003 &   0.193 &   2.0 \\ \hline\hline
$\lambda$ &   0.111 &   0.506 &   3.0 \\ \hline
$\alpha$ &   0.046 &   0.634 &   3.0 \\ \hline
$\theta$ &     0.697 &   3.004 &   3.0 \\ \hline
 $\nu$ &       0.002 &   0.208 &   2.0 \\ \hline\hline
$\lambda$ &   0.165 &   0.699 &   0.5 \\ \hline
$\alpha$ &   0.022 &   0.112 &   0.5 \\ \hline
$\theta$ &     0.020 &   0.161 &   0.5 \\ \hline
 $\nu$ &       0.000 &   0.309 &   3.0 \\ \hline \hline
$\lambda$ &   0.244 &   0.926 &   1.0 \\ \hline
$\alpha$ &   0.027 &   0.120 &   0.5 \\ \hline
$\theta$ &     0.019 &   0.162 &   0.5 \\ \hline
 $\nu$ &       0.004 &   0.325 &   3.0 \\ \hline\hline
$\lambda$ &   0.644 &   3.019 &   3.0 \\ \hline
$\alpha$ &   0.042 &   0.147 &   0.5 \\ \hline
$\theta$ &     0.004 &   0.166 &   0.5 \\ \hline
 $\nu$ &      0.063 &   0.345 &   3.0 \\ \hline\hline
$\lambda$ &   0.049 &   0.221 &   0.5 \\ \hline
$\alpha$ &   0.061 &   0.260 &   1.0 \\ \hline
$\theta$ &     0.023 &   0.177 &   0.5 \\ \hline
 $\nu$ &      $-$0.021 &   0.318 &   3.0 \\ \hline\hline
$\lambda$ &   0.067 &   0.401 &   1.0 \\ \hline
$\alpha$ &   0.069 &   0.293 &   1.0 \\ \hline 
$\theta$ &    0.009 &   0.166 &   0.5 \\ \hline 
 $\nu$ &     0.023 &   0.328 &   3.0 \\ \hline\hline
$\lambda$ &   0.365 &   1.527 &   3.0 \\ \hline
$\alpha$ &   0.050 &   0.253 &   1.0 \\ \hline
$\theta$ &     0.033 &   0.190 &   0.5 \\ \hline
 $\nu$ &      $-$0.007 &   0.330 &   3.0 \\ \hline\hline
$\lambda$ &   0.011 &   0.061 &   0.5 \\ \hline
$\alpha$ &   0.171 &   0.821 &   3.0 \\ \hline
$\theta$ &     0.019 &   0.164 &   0.5 \\ \hline
 $\nu$ &      $-$0.015 &   0.329 &   3.0 \\ \hline\hline
$\lambda$ &   0.019 &   0.121 &   1.0 \\ \hline
$\alpha$ &   0.188 &   0.779 &   3.0 \\ \hline
$\theta$ &     0.022 &   0.169 &   0.5 \\ \hline
 $\nu$ &      $-$0.027 &   0.309 &   3.0 \\ \hline \hline
$\lambda$ &   0.026 &   0.337 &   3.0 \\ \hline
$\alpha$ &   0.219 &   0.788 &   3.0 \\ \hline
$\theta$ &     0.000 &   0.157 &   0.5 \\ \hline
 $\nu$ &      $-$0.003 &   0.307 &   3.0 \\ \hline\hline
$\lambda$ &   0.295 &   1.045 &   0.5 \\ \hline
$\alpha$ &      0.014 &   0.106 &   0.5 \\ \hline
$\theta$ &     0.111 &   0.482 &   1.0 \\ \hline
 $\nu$ &       0.008 &   0.309 &   3.0 \\ \hline\hline
$\lambda$ &   0.415 &   1.777 &   1.0 \\ \hline
$\alpha$ &   0.014 &   0.106 &   0.5 \\ \hline
$\theta$ &     0.070 &   0.430 &   1.0 \\ \hline
 $\nu$ &       0.008 &   0.330 &   3.0 \\ \hline\hline
$\lambda$ &   1.186 &   5.582 &   3.0 \\ \hline
$\alpha$ &   0.020 &   0.114 &   0.5 \\ \hline
$\theta$ &     0.083 &   0.426 &   1.0 \\ \hline
 $\nu$ &       0.021 &   0.326 &   3.0 \\ \hline\hline
$\lambda$ &   0.038 &   0.221 &   0.5 \\ \hline
$\alpha$ &   0.059 &   0.217 &   1.0 \\ \hline
$\theta$ &     0.049 &   0.402 &   1.0 \\ \hline
 $\nu$ &       0.011 &   0.327 &   3.0 \\ \hline\hline
$\lambda$ &   0.122 &   0.495 &   1.0 \\ \hline
$\alpha$ &   0.042 &   0.210 &   1.0 \\ \hline
$\theta$ &     0.073 &   0.407 &   1.0 \\ \hline
 $\nu$ &      $-$0.039 &   0.348 &   3.0 \\ \hline\hline
$\lambda$ &   0.230 &   1.226 &   3.0 \\ \hline
$\alpha$ &   0.048 &   0.216 &   1.0 \\ \hline 
$\theta$ &    0.045 &   0.383 &   1.0 \\ \hline
 $\nu$ &     $-$0.011 &   0.336 &   3.0 \\ \hline \hline
$\lambda$ &   0.007 &   0.057 &   0.5 \\ \hline
$\alpha$ &   0.169 &   0.715 &   3.0 \\ \hline
$\theta$ &     0.049 &   0.391 &   1.0 \\ \hline
 $\nu$ &       0.013 &   0.331 &   3.0 \\ \hline\hline
$\lambda$ &   0.041 &   0.133 &   1.0 \\ \hline
$\alpha$ &   0.005 &   0.633 &   3.0 \\ \hline 
$\theta$ &    0.168 &   0.510 &   1.0 \\ \hline
 $\nu$ &     $-$0.007 &   0.333 &   3.0 \\ \hline\hline
$\lambda$ &   0.057 &   0.357 &   3.0 \\ \hline
$\alpha$ &   0.107 &   0.657 &   3.0 \\ \hline
$\theta$ &     0.072 &   0.388 &   1.0 \\ \hline
 $\nu$ &      $-$0.007 &   0.326 &   3.0 \\ \hline\hline
$\lambda$ &   0.329 &   1.211 &   0.5 \\ \hline
$\alpha$ &   0.009 &   0.099 &   0.5 \\ \hline
$\theta$ &     0.523 &   2.274 &   3.0 \\ \hline
 $\nu$ &      $-$0.014 &   0.326 &   3.0 \\ \hline\hline
$\lambda$ &   0.561 &   2.042 &   1.0 \\ \hline 
$\alpha$ &  0.008 &   0.096 &   0.5 \\ \hline
$\theta$ &    0.572 &   2.067 &   3.0 \\ \hline
  $\nu$ &    $-$0.025 &   0.335 &   3.0 \\ \hline\hline
$\lambda$ &   1.438 &   9.755 &   3.0 \\ \hline
$\alpha$ &   0.030 &   0.109 &   0.5 \\ \hline 
$\theta$ &    0.369 &   2.425 &   3.0 \\ \hline
 $\nu$ &     $-$0.024 &   0.347 &   3.0 \\ \hline\hline
$\lambda$ &   0.053 &   0.261 &   0.5 \\ \hline
$\alpha$ &   0.035 &   0.204 &   1.0 \\ \hline
$\theta$ &     0.339 &   1.702 &   3.0 \\ \hline
 $\nu$ &       0.025 &   0.333 &   3.0 \\ \hline \hline
$\lambda$ &   0.161 &   0.541 &   1.0 \\ \hline
$\alpha$ &   0.016 &   0.202 &   1.0 \\ \hline
$\theta$ &     0.564 &   2.098 &   3.0 \\ \hline
 $\nu$ &      $-$0.010 &   0.300 &   3.0 \\ \hline\hline
$\lambda$ &   0.524 &   2.211 &   3.0 \\ \hline
$\alpha$ &   0.025 &   0.206 &   1.0 \\ \hline 
$\theta$ &    0.556 &   2.404 &   3.0 \\ \hline
 $\nu$ &     $-$0.002 &   0.343 &   3.0 \\ \hline\hline
$\lambda$ &   0.004 &   0.065 &   0.5 \\ \hline
$\alpha$ &   0.146 &   0.612 &   3.0 \\ \hline
$\theta$ &     0.270 &   1.736 &   3.0 \\ \hline
 $\nu$ &       0.026 &   0.326 &   3.0 \\ \hline\hline
$\lambda$ &   0.008 &   0.149 &   1.0 \\ \hline 
$\alpha$ &  0.153 &   0.607 &   3.0 \\ \hline 
$\theta$ &   0.338 &   2.340 &   3.0 \\ \hline 
 $\nu$ &    0.016 &   0.327 &   3.0 \\ \hline\hline
$\lambda$ &   0.037 &   0.402 &   3.0 \\ \hline 
$\alpha$ &  0.137 &   0.634 &   3.0 \\ \hline 
$\theta$ &   0.316 &   1.881 &   3.0 \\ \hline
 $\nu$ &    $-$0.014 &   0.303 &   3.0 \\ \hline\hline
 \caption{Bias and MSE Table for $n=100$}
 \label{biasmse100}
\end{longtable}

\clearpage

\begin{longtable}{|c|r|r|c|}
\hline
\multicolumn{4}{|c|}{n=250}\\ \hline
Parameter & Bias & MSE & True Value \\ \hline  
$\lambda$ &   0.032 &   0.203 &   0.5 \\ \hline
$\alpha$ &   0.012 &   0.064 &   0.5 \\ \hline
$\theta$ &     0.001 &   0.088 &   0.5 \\ \hline
 $\nu$ &       0.011 &   0.019 &   1.0 \\ \hline\hline
$\lambda$ &   0.069 &   0.416 &   1.0 \\ \hline 
$\alpha$ &  0.004 &   0.055 &   0.5 \\ \hline
$\theta$ &    0.007 &   0.085 &   0.5 \\ \hline
  $\nu$ &     0.011 &   0.020 &   1.0 \\ \hline\hline
$\lambda$ &   0.192 &   1.047 &   3.0 \\ \hline
$\alpha$ &   0.002 &   0.054 &   0.5 \\ \hline
$\theta$ &     0.006 &   0.074 &   0.5 \\ \hline
  $\nu$ &      0.013 &   0.023 &   1.0 \\ \hline\hline
$\lambda$ &  $-$0.001 &   0.080 &   0.5 \\ \hline
$\alpha$ &   0.033 &   0.132 &   1.0 \\ \hline
$\theta$ &    $-$0.007 &   0.082 &   0.5 \\ \hline
  $\nu$ &     0.013 &   0.023 &   1.0 \\ \hline\hline
$\lambda$ &   0.009 &   0.184 &   1.0 \\ \hline
$\alpha$ &   0.028 &   0.140 &   1.0 \\ \hline
$\theta$ &     0.001 &   0.094 &   0.5 \\ \hline
  $\nu$ &      0.013 &   0.021 &   1.0 \\ \hline\hline
$\lambda$ &   0.033 &   0.443 &   3.0 \\ \hline
$\alpha$ &   0.019 &   0.114 &   1.0 \\ \hline
$\theta$ &     0.000 &   0.075 &   0.5 \\ \hline
 $\nu$ &       0.010 &   0.018 &   1.0 \\ \hline\hline
$\lambda$ &   0.003 &   0.028 &   0.5 \\ \hline
$\alpha$ &   0.025 &   0.350 &   3.0 \\ \hline
$\theta$ &     0.008 &   0.086 &   0.5 \\ \hline
  $\nu$ &      0.014 &   0.024 &   1.0 \\ \hline   \hline
$\lambda$ &   0.007 &   0.061 &   1.0 \\ \hline
$\alpha$ &   0.032 &   0.425 &   3.0 \\ \hline
$\theta$ &     0.010 &   0.093 &   0.5 \\ \hline
  $\nu$ &      0.012 &   0.020 &   1.0 \\ \hline\hline
$\lambda$ &   0.004 &   0.182 &   3.0 \\ \hline
$\alpha$ &   0.069 &   0.396 &   3.0 \\ \hline
$\theta$ &     0.003 &   0.092 &   0.5 \\ \hline
  $\nu$ &      0.011 &   0.018 &   1.0 \\ \hline\hline
$\lambda$ &   0.025 &   0.201 &   0.5 \\ \hline
$\alpha$ &   0.011 &   0.059 &   0.5 \\ \hline
$\theta$ &     0.005 &   0.189 &   1.0 \\ \hline
  $\nu$ &      0.011 &   0.019 &   1.0 \\ \hline\hline
$\lambda$ &   0.112 &   0.401 &   1.0 \\ \hline
$\alpha$ &   0.000 &   0.051 &   0.5 \\ \hline
$\theta$ &     0.040 &   0.178 &   1.0 \\ \hline
  $\nu$ &      0.011 &   0.022 &   1.0 \\ \hline\hline
$\lambda$ &   0.324 &   1.504 &   3.0 \\ \hline
$\alpha$ &   0.004 &   0.055 &   0.5 \\ \hline
$\theta$ &     0.027 &   0.206 &   1.0 \\ \hline
  $\nu$ &      0.012 &   0.020 &   1.0 \\ \hline\hline
$\lambda$ &   0.016 &   0.100 &   0.5 \\ \hline
$\alpha$ &   0.010 &   0.120 &   1.0 \\ \hline
$\theta$ &     0.023 &   0.202 &   1.0 \\ \hline
 $\nu$ &       0.012 &   0.019 &   1.0 \\ \hline\hline
$\lambda$ &   0.014 &   0.199 &   1.0 \\ \hline
$\alpha$ &   0.018 &   0.113 &   1.0 \\ \hline 
$\theta$ &    0.010 &   0.205 &   1.0 \\ \hline
  $\nu$ &     0.011 &   0.019 &   1.0 \\ \hline\hline
$\lambda$ &   0.130 &   0.661 &   3.0 \\ \hline
 $\alpha$ &  0.001 &   0.113 &   1.0 \\ \hline
 $\theta$ &     0.034 &   0.218 &   1.0 \\ \hline
  $\nu$ &      0.013 &   0.021 &   1.0 \\ \hline\hline
$\lambda$ &   0.003 &   0.031 &   0.5 \\ \hline
$\alpha$ &   0.031 &   0.345 &   3.0 \\ \hline
 $\theta$ &   0.024 &   0.215 &   1.0 \\ \hline
 $\nu$ &      0.010 &   0.018 &   1.0 \\ \hline\hline
$\lambda$ &   0.008 &   0.059 &   1.0 \\ \hline
$\alpha$ &   0.019 &   0.345 &   3.0 \\ \hline 
 $\theta$ &   0.030 &   0.206 &   1.0 \\ \hline
  $\nu$ &     0.013 &   0.021 &   1.0 \\ \hline\hline
$\lambda$ &   0.018 &   0.171 &   3.0 \\ \hline
$\alpha$ &   0.042 &   0.355 &   3.0 \\ \hline
 $\theta$ &    0.021 &   0.193 &   1.0 \\ \hline
  $\nu$ &      0.011 &   0.020 &   1.0 \\ \hline\hline
$\lambda$ &   0.029 &   0.229 &   0.5 \\ \hline
$\alpha$ &   0.005 &   0.054 &   0.5 \\ \hline
 $\theta$ &    0.031 &   0.689 &   3.0 \\ \hline
  $\nu$ &      0.011 &   0.017 &   1.0 \\ \hline\hline
$\lambda$ &   0.173 &   0.638 &   1.0 \\ \hline
$\alpha$ &  $-$0.003 &   0.049 &   0.5 \\ \hline
 $\theta$ &   0.221 &   0.892 &   3.0 \\ \hline
  $\nu$ &     0.011 &   0.018 &   1.0 \\ \hline\hline
$\lambda$ &   0.199 &   1.598 &   3.0 \\ \hline
$\alpha$ &   0.008 &   0.050 &   0.5 \\ \hline
 $\theta$ &    0.047 &   0.732 &   3.0 \\ \hline
 $\nu$ &       0.011 &   0.019 &   1.0 \\ \hline   \hline
$\lambda$ &   0.011 &   0.094 &   0.5 \\ \hline
$\alpha$ &   0.006 &   0.094 &   1.0 \\ \hline
  $\theta$ &   0.077 &   0.685 &   3.0 \\ \hline
  $\nu$ &      0.014 &   0.023 &   1.0 \\ \hline\hline
$\lambda$ &   0.011 &   0.172 &   1.0 \\ \hline
$\alpha$ &   0.016 &   0.105 &   1.0 \\ \hline
 $\theta$ &    0.025 &   0.623 &   3.0 \\ \hline
  $\nu$ &      0.013 &   0.022 &   1.0 \\ \hline\hline
$\lambda$ &   0.115 &   0.614 &   3.0 \\ \hline
$\alpha$ &   0.002 &   0.093 &   1.0 \\ \hline
 $\theta$ &    0.160 &   0.745 &   3.0 \\ \hline
  $\nu$ &      0.013 &   0.020 &   1.0 \\ \hline\hline
$\lambda$ &   0.000 &   0.032 &   0.5 \\ \hline
$\alpha$ &   0.050 &   0.321 &   3.0 \\ \hline
 $\theta$ &    0.042 &   0.732 &   3.0 \\ \hline
  $\nu$ &      0.011 &   0.019 &   1.0 \\ \hline\hline
$\lambda$ &   0.001 &   0.059 &   1.0 \\ \hline
$\alpha$ &   0.039 &   0.304 &   3.0 \\ \hline
 $\theta$ &    0.044 &   0.672 &   3.0 \\ \hline
  $\nu$ &      0.015 &   0.023 &   1.0 \\ \hline\hline
$\lambda$ &   0.035 &   0.216 &   3.0 \\ \hline
$\alpha$ &   0.002 &   0.328 &   3.0 \\ \hline
 $\theta$ &    0.179 &   0.871 &   3.0 \\ \hline
  $\nu$ &      0.014 &   0.021 &   1.0 \\ \hline\hline
$\lambda$ &   0.059 &   0.240 &   0.5 \\ \hline
$\alpha$ &   0.013 &   0.071 &   0.5 \\ \hline
 $\theta$ &    0.005 &   0.099 &   0.5 \\ \hline
  $\nu$ &      0.002 &   0.129 &   2.0 \\ \hline  \hline
$\lambda$ &   0.015 &   0.382 &   1.0 \\ \hline
$\alpha$ &   0.018 &   0.071 &   0.5 \\ \hline
 $\theta$ &   $-$0.008 &   0.086 &   0.5 \\ \hline
  $\nu$ &     0.003 &   0.130 &   2.0 \\ \hline\hline
$\lambda$ &   0.483 &   2.074 &   3.0 \\ \hline
$\alpha$ &   0.003 &   0.067 &   0.5 \\ \hline
 $\theta$ &    0.018 &   0.102 &   0.5 \\ \hline
  $\nu$ &      0.002 &   0.128 &   2.0 \\ \hline\hline
$\lambda$ &   0.021 &   0.107 &   0.5 \\ \hline
$\alpha$ &   0.012 &   0.137 &   1.0 \\ \hline
 $\theta$ &    0.009 &   0.101 &   0.5 \\ \hline
  $\nu$ &     $-$0.016 &   0.118 &   2.0 \\ \hline\hline
$\lambda$ &   0.019 &   0.203 &   1.0 \\ \hline
$\alpha$ &   0.031 &   0.128 &   1.0 \\ \hline
 $\theta$ &    0.001 &   0.095 &   0.5 \\ \hline
  $\nu$ &     $-$0.016 &   0.132 &   2.0 \\ \hline\hline
$\lambda$ &   0.157 &   0.700 &   3.0 \\ \hline
$\alpha$ &   0.006 &   0.139 &   1.0 \\ \hline
 $\theta$ &    0.013 &   0.097 &   0.5 \\ \hline
  $\nu$ &     $-$0.002 &   0.131 &   2.0 \\ \hline\hline
$\lambda$ &   0.005 &   0.033 &   0.5 \\ \hline
$\alpha$ &   0.016 &   0.399 &   3.0 \\ \hline
 $\theta$ &    0.020 &   0.095 &   0.5 \\ \hline
 $\nu$ &      $-$0.008 &   0.122 &   2.0 \\ \hline\hline
$\lambda$ &  $-$0.002 &   0.063 &   1.0 \\ \hline
$\alpha$ &   0.103 &   0.428 &   3.0 \\ \hline
 $\theta$ &   $-$0.002 &   0.091 &   0.5 \\ \hline
 $\nu$ &      0.004 &   0.125 &   2.0 \\ \hline \hline
$\lambda$ &   0.015 &   0.206 &   3.0 \\ \hline
$\alpha$ &   0.073 &   0.437 &   3.0 \\ \hline
 $\theta$ &    0.003 &   0.098 &   0.5 \\ \hline
 $\nu$ &       0.001 &   0.131 &   2.0 \\ \hline\hline
$\lambda$ &   0.079 &   0.275 &   0.5 \\ \hline
$\alpha$ &   0.001 &   0.056 &   0.5 \\ \hline
 $\theta$ &    0.046 &   0.222 &   1.0 \\ \hline
 $\nu$ &      $-$0.002 &   0.127 &   2.0 \\ \hline\hline
$\lambda$ &   0.089 &   0.462 &   1.0 \\ \hline
$\alpha$ &   0.009 &   0.064 &   0.5 \\ \hline
 $\theta$ &    0.021 &   0.216 &   1.0 \\ \hline
 $\nu$ &       0.004 &   0.126 &   2.0 \\ \hline\hline
$\lambda$ &   0.254 &   1.286 &   3.0 \\ \hline
$\alpha$ &   0.005 &   0.056 &   0.5 \\ \hline
 $\theta$ &    0.021 &   0.190 &   1.0 \\ \hline
  $\nu$ &      0.010 &   0.140 &   2.0 \\ \hline\hline
$\lambda$ &   0.005 &   0.096 &   0.5 \\ \hline
$\alpha$ &   0.023 &   0.117 &   1.0 \\ \hline
 $\theta$ &    0.005 &   0.196 &   1.0 \\ \hline
  $\nu$ &      0.005 &   0.114 &   2.0 \\ \hline\hline
$\lambda$ &   0.021 &   0.205 &   1.0 \\ \hline
$\alpha$ &   0.021 &   0.129 &   1.0 \\ \hline
  $\theta$ &   0.015 &   0.213 &   1.0 \\ \hline
  $\nu$ &      0.013 &   0.133 &   2.0 \\ \hline\hline
$\lambda$ &   0.105 &   0.674 &   3.0 \\ \hline
$\alpha$ &   0.020 &   0.135 &   1.0 \\ \hline 
 $\theta$ &   0.022 &   0.224 &   1.0 \\ \hline
 $\nu$ &     $-$0.009 &   0.126 &   2.0 \\ \hline  \hline
$\lambda$ &   0.002 &   0.034 &   0.5 \\ \hline
$\alpha$ &   0.068 &   0.404 &   3.0 \\ \hline
 $\theta$ &    0.018 &   0.219 &   1.0 \\ \hline
 $\nu$ &      $-$0.015 &   0.126 &   2.0 \\ \hline\hline
$\lambda$ &   0.011 &   0.064 &   1.0 \\ \hline
$\alpha$ &   0.010 &   0.356 &   3.0 \\ \hline
 $\theta$ &    0.038 &   0.209 &   1.0 \\ \hline
 $\nu$ &      $-$0.006 &   0.127 &   2.0 \\ \hline\hline
$\lambda$ &   0.017 &   0.188 &   3.0 \\ \hline
$\alpha$ &   0.033 &   0.360 &   3.0 \\ \hline
 $\theta$ &    0.021 &   0.202 &   1.0 \\ \hline
 $\nu$ &       0.005 &   0.122 &   2.0 \\ \hline\hline
$\lambda$ &   0.095 &   0.379 &   0.5 \\ \hline
$\alpha$ &   0.007 &   0.065 &   0.5 \\ \hline 
 $\theta$ &   0.153 &   1.007 &   3.0 \\ \hline
 $\nu$ &     $-$0.005 &   0.133 &   2.0 \\ \hline\hline
$\lambda$ &   0.151 &   0.830 &   1.0 \\ \hline 
$\alpha$ &  0.009 &   0.063 &   0.5 \\ \hline 
$\theta$ &   0.121 &   1.028 &   3.0 \\ \hline
 $\nu$ &     0.013 &   0.133 &   2.0 \\ \hline\hline
$\lambda$ &   0.554 &   2.378 &   3.0 \\ \hline 
$\alpha$ &  0.007 &   0.067 &   0.5 \\ \hline
 $\theta$ &   0.184 &   1.017 &   3.0 \\ \hline
 $\nu$ &     $-$0.013 &   0.126 &   2.0 \\ \hline\hline
$\lambda$ &   0.011 &   0.121 &   0.5 \\ \hline
$\alpha$ &   0.021 &   0.120 &   1.0 \\ \hline
 $\theta$ &    0.067 &   0.889 &   3.0 \\ \hline
  $\nu$ &     $-$0.004 &   0.127 &   2.0 \\ \hline \hline  
$\lambda$ &   0.041 &   0.252 &   1.0 \\ \hline
$\alpha$ &   0.013 &   0.125 &   1.0 \\ \hline
 $\theta$ &    0.157 &   0.950 &   3.0 \\ \hline
 $\nu$ &       0.008 &   0.117 &   2.0 \\ \hline\hline
$\lambda$ &   0.133 &   0.867 &   3.0 \\ \hline
$\alpha$ &   0.012 &   0.126 &   1.0 \\ \hline 
$\theta$ &    0.139 &   0.989 &   3.0 \\ \hline
 $\nu$ &      0.012 &   0.122 &   2.0 \\ \hline\hline
$\lambda$ &   0.002 &   0.034 &   0.5 \\ \hline
$\alpha$ &   0.042 &   0.336 &   3.0 \\ \hline
$\theta$ &     0.105 &   0.807 &   3.0 \\ \hline
  $\nu$ &     $-$0.005 &   0.128 &   2.0 \\ \hline\hline
$\lambda$ &   0.012 &   0.075 &   1.0 \\ \hline
$\alpha$ &   0.009 &   0.367 &   3.0 \\ \hline
$\theta$ &     0.179 &   0.893 &   3.0 \\ \hline
  $\nu$ &      0.013 &   0.130 &   2.0 \\ \hline\hline
$\lambda$ &   0.019 &   0.227 &   3.0 \\ \hline
$\alpha$ &   0.050 &   0.373 &   3.0 \\ \hline
 $\theta$ &    0.115 &   0.878 &   3.0 \\ \hline
  $\nu$ &     $-$0.010 &   0.124 &   2.0 \\ \hline\hline
$\lambda$ &   0.043 &   0.229 &   0.5 \\ \hline
$\alpha$ &   0.013 &   0.066 &   0.5 \\ \hline
 $\theta$ &    0.002 &   0.091 &   0.5 \\ \hline
 $\nu$ &      $-$0.015 &   0.199 &   3.0 \\ \hline\hline
$\lambda$ &   0.129 &   0.567 &   1.0 \\ \hline
$\alpha$ &   0.008 &   0.070 &   0.5 \\ \hline
 $\theta$ &    0.010 &   0.107 &   0.5 \\ \hline
 $\nu$ &       0.003 &   0.192 &   3.0 \\ \hline 
$\lambda$ &   0.257 &   1.533 &   3.0 \\ \hline \hline
$\alpha$ &   0.009 &   0.065 &   0.5 \\ \hline
 $\theta$ &    0.002 &   0.091 &   0.5 \\ \hline
 $\nu$ &       0.023 &   0.205 &   3.0 \\ \hline\hline
$\lambda$ &   0.013 &   0.094 &   0.5 \\ \hline
$\alpha$ &   0.016 &   0.128 &   1.0 \\ \hline
 $\theta$ &    0.009 &   0.090 &   0.5 \\ \hline
 $\nu$ &       0.011 &   0.207 &   3.0 \\ \hline\hline
$\lambda$ &   0.045 &   0.215 &   1.0 \\ \hline
$\alpha$ &   0.016 &   0.130 &   1.0 \\ \hline
 $\theta$ &    0.007 &   0.095 &   0.5 \\ \hline
  $\nu$ &     $-$0.028 &   0.212 &   3.0 \\ \hline\hline
$\lambda$ &   0.100 &   0.657 &   3.0 \\ \hline
$\alpha$ &   0.025 &   0.140 &   1.0 \\ \hline
 $\theta$ &    0.008 &   0.103 &   0.5 \\ \hline
 $\nu$ &      $-$0.004 &   0.193 &   3.0 \\ \hline\hline
$\lambda$ &   0.003 &   0.034 &   0.5 \\ \hline
$\alpha$ &   0.045 &   0.407 &   3.0 \\ \hline
$\theta$ &     0.007 &   0.095 &   0.5 \\ \hline
 $\nu$ &       0.000 &   0.213 &   3.0 \\ \hline\hline
$\lambda$ &   0.008 &   0.073 &   1.0 \\ \hline
$\alpha$ &   0.054 &   0.438 &   3.0 \\ \hline
$\theta$ &     0.014 &   0.106 &   0.5 \\ \hline
  $\nu$ &      0.008 &   0.209 &   3.0 \\ \hline\hline
$\lambda$ &   0.018 &   0.202 &   3.0 \\ \hline
$\alpha$ &   0.050 &   0.411 &   3.0 \\ \hline 
 $\theta$ &   0.007 &   0.101 &   0.5 \\ \hline
  $\nu$ &    $-$0.011 &   0.193 &   3.0 \\ \hline   \hline
$\lambda$ &   0.056 &   0.274 &   0.5 \\ \hline
$\alpha$ &   0.009 &   0.064 &   0.5 \\ \hline
$\theta$ &     0.017 &   0.231 &   1.0 \\ \hline
  $\nu$ &      0.011 &   0.201 &   3.0 \\ \hline\hline
$\lambda$ &   0.125 &   0.493 &   1.0 \\ \hline
$\alpha$ &   0.004 &   0.062 &   0.5 \\ \hline
$\theta$ &     0.028 &   0.210 &   1.0 \\ \hline
  $\nu$ &     $-$0.022 &   0.206 &   3.0 \\ \hline\hline
$\lambda$ &   0.259 &   1.528 &   3.0 \\ \hline
$\alpha$ &   0.011 &   0.060 &   0.5 \\ \hline
$\theta$ &     0.010 &   0.213 &   1.0 \\ \hline
  $\nu$ &     $-$0.012 &   0.204 &   3.0 \\ \hline\hline
$\lambda$ &   0.017 &   0.102 &   0.5 \\ \hline
$\alpha$ &   0.014 &   0.124 &   1.0 \\ \hline
$\theta$ &     0.029 &   0.210 &   1.0 \\ \hline
  $\nu$ &      0.001 &   0.203 &   3.0 \\ \hline\hline
$\lambda$ &   0.039 &   0.217 &   1.0 \\ \hline
$\alpha$ &   0.010 &   0.124 &   1.0 \\ \hline
$\theta$ &     0.037 &   0.222 &   1.0 \\ \hline
  $\nu$ &      0.012 &   0.193 &   3.0 \\ \hline\hline
$\lambda$ &   0.163 &   0.700 &   3.0 \\ \hline
$\alpha$ &   0.011 &   0.129 &   1.0 \\ \hline
 $\theta$ &    0.049 &   0.239 &   1.0 \\ \hline
  $\nu$ &     $-$0.018 &   0.197 &   3.0 \\ \hline\hline
$\lambda$ &   0.004 &   0.036 &   0.5 \\ \hline
$\alpha$ &   0.038 &   0.380 &   3.0 \\ \hline
 $\theta$ &    0.025 &   0.223 &   1.0 \\ \hline
  $\nu$ &      0.013 &   0.212 &   3.0 \\ \hline    \hline
$\lambda$ &   0.006 &   0.068 &   1.0 \\ \hline
$\alpha$ &   0.050 &   0.356 &   3.0 \\ \hline
 $\theta$ &    0.017 &   0.217 &   1.0 \\ \hline
  $\nu$ &     $-$0.019 &   0.211 &   3.0 \\ \hline\hline
$\lambda$ &   0.026 &   0.211 &   3.0 \\ \hline
$\alpha$ &   0.042 &   0.376 &   3.0 \\ \hline
 $\theta$ &    0.030 &   0.230 &   1.0 \\ \hline
  $\nu$ &     $-$0.017 &   0.196 &   3.0 \\ \hline\hline
$\lambda$ &   0.035 &   0.279 &   0.5 \\ \hline
$\alpha$ &   0.013 &   0.059 &   0.5 \\ \hline
 $\theta$ &    0.038 &   0.846 &   3.0 \\ \hline
  $\nu$ &     $-$0.008 &   0.217 &   3.0 \\ \hline\hline
$\lambda$ &   0.073 &   0.549 &   1.0 \\ \hline
$\alpha$ &   0.013 &   0.058 &   0.5 \\ \hline
$\theta$ &     0.052 &   0.816 &   3.0 \\ \hline
  $\nu$ &     $-$0.008 &   0.201 &   3.0 \\ \hline\hline
$\lambda$ &   0.491 &   2.629 &   3.0 \\ \hline
$\alpha$ &   0.005 &   0.058 &   0.5 \\ \hline
$\theta$ &     0.164 &   0.984 &   3.0 \\ \hline
  $\nu$ &      0.009 &   0.208 &   3.0 \\ \hline\hline
$\lambda$ &   0.027 &   0.136 &   0.5 \\ \hline
$\alpha$ &   0.008 &   0.123 &   1.0 \\ \hline
 $\theta$ &    0.181 &   0.975 &   3.0 \\ \hline
   $\nu$ &    $-$0.002 &   0.219 &   3.0 \\ \hline\hline
$\lambda$ &   0.071 &   0.261 &   1.0 \\ \hline
$\alpha$ &  $-$0.001 &   0.125 &   1.0 \\ \hline
$\theta$ &    0.259 &   1.020 &   3.0 \\ \hline
  $\nu$ &    $-$0.004 &   0.198 &   3.0 \\ \hline    \hline
$\lambda$ &   0.064 &   0.681 &   3.0 \\ \hline
$\alpha$ &   0.021 &   0.126 &   1.0 \\ \hline
$\theta$ &     0.066 &   0.840 &   3.0 \\ \hline
  $\nu$ &      0.003 &   0.213 &   3.0 \\ \hline\hline
$\lambda$ &   0.009 &   0.043 &   0.5 \\ \hline
$\alpha$ &   0.007 &   0.388 &   3.0 \\ \hline
$\theta$ &     0.271 &   1.112 &   3.0 \\ \hline
  $\nu$ &     $-$0.019 &   0.210 &   3.0 \\ \hline\hline
$\lambda$ &   0.000 &   0.080 &   1.0 \\ \hline
$\alpha$ &   0.085 &   0.397 &   3.0 \\ \hline
$\theta$ &     0.088 &   0.983 &   3.0 \\ \hline
  $\nu$ &      0.000 &   0.192 &   3.0 \\ \hline\hline
$\lambda$ &   0.031 &   0.247 &   3.0 \\ \hline
$\alpha$ &   0.035 &   0.403 &   3.0 \\ \hline
$\theta$ &     0.196 &   1.004 &   3.0 \\ \hline
  $\nu$ &      0.003 &   0.202 &   3.0 \\ \hline\hline
  \caption{Bias and MSE Table for $n=250$}
  \label{biasmse250}
\end{longtable}

\clearpage
\section{Confidence Regions Tables}
 The following tables provide the estimated coverage probabilities for the each of the confidence regions discussed in Chapter 3. The first column provides the estimated coverage probability generated from the Mahalanobis distance using the observed information matrix in (\ref{infoconf}). The second column provides the estimated coverage probability from the Mahalanobis distance using the bootstrap estimated covariance matrix in (\ref{bscovariance}), and then using the confidence region procedure found in (\ref{bsconfreg}). The third column provides the coverage probabilities for the likelihood ratio test found in (\ref{LRconf}). The fourth through seventh columns provide the values of the parameters from the model found in (\ref{mvewcdf}). In some cases, the Mahalanobis distance based on the information matrix was negative definite. If this happened, the sample was not used to compute the coverage probabilities. The eighth column provides the number of Mahalanobis distance based on the information matrix that were used to compute the coverage probability (with a maximum of 250).  
 \clearpage
\begin{longtable}{|r|r|r|c|c|c|c|c|}
\hline
\multicolumn{8}{|c|}{$n=50$}\\ \hline
Information & Bootstrap & Ratio & $\lambda$ & $\alpha$ & $\theta$ & $\nu$ & $N$\\ \hline
  0.706 &  0.892 &  0.968 &  0.5 &  0.5 &   0.5 &  1.0 & 248     \\ \hline
  0.778 &   0.940 &   0.940 &   1.0 &   0.5 &   0.5 &   1.0 & 248\\ \hline
  0.818 &   0.932 &   0.960 &   3.0 &   0.5 &   0.5 &   1.0 & 247\\ \hline \hline
  0.776 &   0.916 &   0.960 &   0.5 &   1.0 &   0.5 &   1.0 & 250\\ \hline
  0.775 &   0.936 &   0.972 &   1.0 &   1.0 &   0.5 &   1.0 & 249\\ \hline
  0.840 &   0.932 &   0.964 &   3.0 &   1.0 &   0.5 &   1.0 & 250\\ \hline \hline
  0.855 &   0.940 &   0.960 &   0.5 &   3.0 &   0.5 &   1.0 & 248\\ \hline
  0.811 &   0.924 &   0.956 &   1.0 &   3.0 &   0.5 &   1.0 & 249\\ \hline
  0.804 &   0.936 &   0.960 &   3.0 &   3.0 &   0.5 &   1.0 & 250\\ \hline \hline
  0.736 &   0.924 &   0.960 &   0.5 &   0.5 &   1.0 &   1.0 & 250\\ \hline
  0.776 &   0.956 &   0.964 &   1.0 &   0.5 &   1.0 &   1.0 & 250\\ \hline
  0.812 &   0.932 &   0.976 &   3.0 &   0.5 &   1.0 &   1.0 & 250\\ \hline \hline
  0.784 &   0.952 &   0.984 &   0.5 &   1.0 &   1.0 &   1.0 & 250\\ \hline
  0.752 &   0.936 &   0.980 &   1.0 &   1.0 &   1.0 &   1.0 & 250\\ \hline
  0.780 &   0.928 &   0.948 &   3.0 &   1.0 &   1.0 &   1.0 & 250\\ \hline \hline
  0.784 &   0.924 &   0.948 &   0.5 &   3.0 &   1.0 &   1.0 & 250\\ \hline
  0.824 &   0.944 &   0.972 &   1.0 &   3.0 &   1.0 &   1.0 & 250\\ \hline
  0.804 &   0.944 &   0.968 &   3.0 &   3.0 &   1.0 &   1.0 & 250\\ \hline \hline
  0.680 &   0.924 &   0.968 &   0.5 &   0.5 &   3.0 &   1.0 & 250\\ \hline
  0.720 &   0.944 &   0.980 &   1.0 &   0.5 &   3.0 &   1.0 & 250\\ \hline
  0.760 &   0.956 &   0.968 &   3.0 &   0.5 &   3.0 &   1.0 & 250\\ \hline \hline
  0.788 &   0.948 &   0.988 &   0.5 &   1.0 &   3.0 &   1.0 & 250\\ \hline
  0.748 &   0.956 &   0.964 &   1.0 &   1.0 &   3.0 &   1.0 & 250\\ \hline
  0.791 &   0.964 &   0.952 &   3.0 &   1.0 &   3.0 &   1.0 & 249\\ \hline \hline
  0.724 &   0.940 &   0.972 &   0.5 &   3.0 &   3.0 &   1.0 & 250\\ \hline
  0.744 &   0.924 &   0.980 &   1.0 &   3.0 &   3.0 &   1.0 & 250\\ \hline
  0.772 &   0.936 &   0.980 &   3.0 &   3.0 &   3.0 &   1.0 & 250\\ \hline \hline
  0.724 &   0.936 &   0.932 &   0.5 &   0.5 &   0.5 &   2.0 & 250\\ \hline
  0.724 &   0.900 &   0.944 &   1.0 &   0.5 &   0.5 &   2.0 & 250\\ \hline
  0.631 &   0.868 &   0.916 &   3.0 &   0.5 &   0.5 &   2.0 & 249\\ \hline \hline
  0.726 &   0.908 &   0.940 &   0.5 &   1.0 &   0.5 &   2.0 & 248\\ \hline
  0.700 &   0.872 &   0.944 &   1.0 &   1.0 &   0.5 &   2.0 & 247\\ \hline
  0.684 &   0.892 &   0.940 &   3.0 &   1.0 &   0.5 &   2.0 & 250\\ \hline \hline
  0.786 &   0.920 &   0.960 &   0.5 &   3.0 &   0.5 &   2.0 & 248\\ \hline
  0.740 &   0.912 &   0.964 &   1.0 &   3.0 &   0.5 &   2.0 & 250\\ \hline
  0.715 &   0.896 &   0.936 &   3.0 &   3.0 &   0.5 &   2.0 & 249\\ \hline \hline
  0.628 &   0.928 &   0.940 &   0.5 &   0.5 &   1.0 &   2.0 & 250\\ \hline
  0.628 &   0.908 &   0.944 &   1.0 &   0.5 &   1.0 &   2.0 & 250\\ \hline
  0.716 &   0.932 &   0.928 &   3.0 &   0.5 &   1.0 &   2.0 & 250\\ \hline \hline
  0.696 &   0.876 &   0.940 &   0.5 &   1.0 &   1.0 &   2.0 & 250\\ \hline
  0.743 &   0.940 &   0.956 &   1.0 &   1.0 &   1.0 &   2.0 & 249\\ \hline
  0.696 &   0.908 &   0.960 &   3.0 &   1.0 &   1.0 &   2.0 & 250\\ \hline \hline
  0.780 &   0.932 &   0.960 &   0.5 &   3.0 &   1.0 &   2.0 & 250\\ \hline
  0.744 &   0.908 &   0.944 &   1.0 &   3.0 &   1.0 &   2.0 & 250\\ \hline
  0.760 &   0.932 &   0.940 &   3.0 &   3.0 &   1.0 &   2.0 & 250\\ \hline \hline
  0.476 &   0.892 &   0.940 &   0.5 &   0.5 &   3.0 &   2.0 & 250\\ \hline
  0.544 &   0.920 &   0.948 &   1.0 &   0.5 &   3.0 &   2.0 & 250\\ \hline
  0.536 &   0.912 &   0.952 &   3.0 &   0.5 &   3.0 &   2.0 & 250\\ \hline \hline
  0.636 &   0.948 &   0.944 &   0.5 &   1.0 &   3.0 &   2.0 & 250\\ \hline
  0.608 &   0.900 &   0.940 &   1.0 &   1.0 &   3.0 &   2.0 & 250\\ \hline
  0.608 &   0.916 &   0.948 &   3.0 &   1.0 &   3.0 &   2.0 & 250\\ \hline \hline
  0.604 &   0.892 &   0.928 &   0.5 &   3.0 &   3.0 &   2.0 & 250\\ \hline
  0.664 &   0.924 &   0.932 &   1.0 &   3.0 &   3.0 &   2.0 & 250\\ \hline
  0.708 &   0.928 &   0.956 &   3.0 &   3.0 &   3.0 &   2.0 & 250\\ \hline \hline
  0.679 &   0.892 &   0.944 &   0.5 &   0.5 &   0.5 &   3.0 & 249\\ \hline
  0.692 &   0.916 &   0.948 &   1.0 &   0.5 &   0.5 &   3.0 & 247\\ \hline
  0.731 &   0.908 &   0.944 &   0.5 &   3.0 &   0.5 &   3.0 & 249\\ \hline \hline
  0.757 &   0.896 &   0.940 &   1.0 &   3.0 &   0.5 &   3.0 & 247\\ \hline
  0.729 &   0.924 &   0.924 &   3.0 &   3.0 &   0.5 &   3.0 & 247\\ \hline
  0.680 &   0.936 &   0.952 &   0.5 &   0.5 &   1.0 &   3.0 & 250\\ \hline \hline
  0.640 &   0.928 &   0.960 &   1.0 &   0.5 &   1.0 &   3.0 & 250\\ \hline
  0.708 &   0.936 &   0.960 &   3.0 &   0.5 &   1.0 &   3.0 & 250\\ \hline
  0.684 &   0.908 &   0.932 &   0.5 &   1.0 &   1.0 &   3.0 & 250\\ \hline \hline
  0.696 &   0.908 &   0.956 &   1.0 &   1.0 &   1.0 &   3.0 & 250\\ \hline
  0.716 &   0.928 &   0.952 &   3.0 &   1.0 &   1.0 &   3.0 & 250\\ \hline
  0.784 &   0.940 &   0.952 &   0.5 &   3.0 &   1.0 &   3.0 & 250\\ \hline \hline
  0.732 &   0.908 &   0.940 &   1.0 &   3.0 &   1.0 &   3.0 & 250\\ \hline
  0.716 &   0.896 &   0.904 &   3.0 &   3.0 &   1.0 &   3.0 & 250\\ \hline
  0.560 &   0.960 &   0.936 &   0.5 &   0.5 &   3.0 &   3.0 & 250\\ \hline \hline
  0.444 &   0.892 &   0.960 &   1.0 &   0.5 &   3.0 &   3.0 & 250\\ \hline
  0.500 &   0.936 &   0.952 &   3.0 &   0.5 &   3.0 &   3.0 & 250\\ \hline
  0.640 &   0.908 &   0.948 &   0.5 &   1.0 &   3.0 &   3.0 & 250\\ \hline \hline
  0.600 &   0.924 &   0.932 &   1.0 &   1.0 &   3.0 &   3.0 & 250\\ \hline
  0.636 &   0.948 &   0.924 &   3.0 &   1.0 &   3.0 &   3.0 & 250\\ \hline
  0.704 &   0.940 &   0.980 &   0.5 &   3.0 &   3.0 &   3.0 & 250\\ \hline \hline
  0.648 &   0.908 &   0.964 &   1.0 &   3.0 &   3.0 &   3.0 & 250\\ \hline
  0.692 &   0.936 &   0.936 &   3.0 &   3.0 &   3.0 &   3.0 & 250\\ \hline
  0.639 &   0.900 &   0.932 &   3.0 &   0.5 &   0.5 &   3.0 & 249\\ \hline \hline
  0.690 &   0.896 &   0.928 &   0.5 &   1.0 &   0.5 &   3.0 & 248\\ \hline
  0.707 &   0.912 &   0.956 &   1.0 &   1.0 &   0.5 &   3.0 & 249\\ \hline
  0.722 &   0.920 &   0.960 &   3.0 &   1.0 &   0.5 &   3.0 & 248\\ \hline
  \caption{Confidence Region Table for $n=50$}
  \label{confreg50}
  \end{longtable}
  
  \clearpage
\begin{longtable}{|r|r|r|c|c|c|c|c|}
\hline
\multicolumn{8}{|c|}{$n=100$}\\ \hline
Information & Bootstrap & Ratio & $\lambda$ & $\alpha$ & $\theta$ & $\nu$ & $N$\\ \hline
  0.832 &   0.928 &   0.960 &   0.5 &   0.5 &   0.5 &   1.0 & 250\\ \hline
  0.844 &   0.932 &   0.952 &   1.0 &   0.5 &   0.5 &   1.0 & 250\\ \hline
  0.876 &   0.968 &   0.972 &   3.0 &   0.5 &   0.5 &   1.0 & 250\\ \hline \hline
  0.832 &   0.920 &   0.948 &   0.5 &   1.0 &   0.5 &   1.0 & 250\\ \hline
  0.820 &   0.876 &   0.952 &   1.0 &   1.0 &   0.5 &   1.0 & 250\\ \hline
  0.876 &   0.948 &   0.968 &   3.0 &   1.0 &   0.5 &   1.0 & 250\\ \hline \hline
  0.868 &   0.940 &   0.972 &   0.5 &   3.0 &   0.5 &   1.0 & 250\\ \hline
  0.884 &   0.948 &   0.956 &   1.0 &   3.0 &   0.5 &   1.0 & 250\\ \hline
  0.892 &   0.960 &   0.972 &   3.0 &   3.0 &   0.5 &   1.0 & 250\\ \hline \hline
  0.820 &   0.968 &   0.968 &   0.5 &   0.5 &   1.0 &   1.0 & 250\\ \hline
  0.796 &   0.952 &   0.976 &   1.0 &   0.5 &   1.0 &   1.0 & 250\\ \hline
  0.832 &   0.944 &   0.952 &   3.0 &   0.5 &   1.0 &   1.0 & 250\\ \hline \hline
  0.808 &   0.928 &   0.972 &   0.5 &   1.0 &   1.0 &   1.0 & 250\\ \hline
  0.868 &   0.944 &   0.992 &   1.0 &   1.0 &   1.0 &   1.0 & 250\\ \hline
  0.876 &   0.968 &   0.972 &   3.0 &   1.0 &   1.0 &   1.0 & 250\\ \hline \hline
  0.876 &   0.944 &   0.968 &   0.5 &   3.0 &   1.0 &   1.0 & 250\\ \hline
  0.888 &   0.940 &   0.968 &   1.0 &   3.0 &   1.0 &   1.0 & 250\\ \hline
  0.860 &   0.936 &   0.968 &   3.0 &   3.0 &   1.0 &   1.0 & 250\\ \hline \hline
  0.768 &   0.964 &   0.984 &   0.5 &   0.5 &   3.0 &   1.0 & 250\\ \hline
  0.768 &   0.968 &   0.968 &   1.0 &   0.5 &   3.0 &   1.0 & 250\\ \hline
  0.728 &   0.960 &   0.952 &   3.0 &   0.5 &   3.0 &   1.0 & 250\\ \hline \hline
  0.836 &   0.960 &   0.976 &   0.5 &   1.0 &   3.0 &   1.0 & 250\\ \hline
  0.820 &   0.936 &   0.972 &   1.0 &   1.0 &   3.0 &   1.0 & 250\\ \hline
  0.820 &   0.948 &   0.984 &   3.0 &   1.0 &   3.0 &   1.0 & 250\\ \hline \hline
  0.848 &   0.940 &   0.976 &   0.5 &   3.0 &   3.0 &   1.0 & 250\\ \hline
  0.828 &   0.956 &   0.956 &   1.0 &   3.0 &   3.0 &   1.0 & 250\\ \hline
  0.832 &   0.932 &   0.972 &   3.0 &   3.0 &   3.0 &   1.0 & 250\\ \hline \hline
  0.760 &   0.916 &   0.936 &   0.5 &   0.5 &   0.5 &   2.0 & 250\\ \hline
  0.763 &   0.900 &   0.968 &   1.0 &   0.5 &   0.5 &   2.0 & 249\\ \hline
  0.732 &   0.892 &   0.928 &   3.0 &   0.5 &   0.5 &   2.0 & 250\\ \hline \hline
  0.775 &   0.856 &   0.948 &   0.5 &   1.0 &   0.5 &   2.0 & 249\\ \hline
  0.788 &   0.924 &   0.964 &   1.0 &   1.0 &   0.5 &   2.0 & 250\\ \hline
  0.816 &   0.928 &   0.964 &   3.0 &   1.0 &   0.5 &   2.0 & 250\\ \hline \hline
  0.840 &   0.916 &   0.960 &   0.5 &   3.0 &   0.5 &   2.0 & 250\\ \hline
  0.840 &   0.924 &   0.936 &   1.0 &   3.0 &   0.5 &   2.0 & 250\\ \hline
  0.748 &   0.884 &   0.908 &   3.0 &   3.0 &   0.5 &   2.0 & 250\\ \hline \hline
  0.716 &   0.912 &   0.932 &   0.5 &   0.5 &   1.0 &   2.0 & 250\\ \hline
  0.752 &   0.904 &   0.936 &   1.0 &   0.5 &   1.0 &   2.0 & 250\\ \hline
  0.712 &   0.908 &   0.940 &   3.0 &   0.5 &   1.0 &   2.0 & 250\\ \hline \hline
  0.832 &   0.924 &   0.960 &   0.5 &   1.0 &   1.0 &   2.0 & 250\\ \hline
  0.772 &   0.916 &   0.956 &   1.0 &   1.0 &   1.0 &   2.0 & 250\\ \hline
  0.756 &   0.920 &   0.956 &   3.0 &   1.0 &   1.0 &   2.0 & 250\\ \hline \hline
  0.824 &   0.912 &   0.948 &   0.5 &   3.0 &   1.0 &   2.0 & 250\\ \hline
  0.800 &   0.908 &   0.952 &   1.0 &   3.0 &   1.0 &   2.0 & 250\\ \hline
  0.804 &   0.916 &   0.944 &   3.0 &   3.0 &   1.0 &   2.0 & 250\\ \hline \hline
  0.592 &   0.924 &   0.964 &   0.5 &   0.5 &   3.0 &   2.0 & 250\\ \hline
  0.608 &   0.892 &   0.936 &   1.0 &   0.5 &   3.0 &   2.0 & 250\\ \hline
  0.564 &   0.936 &   0.944 &   3.0 &   0.5 &   3.0 &   2.0 & 250\\ \hline \hline
  0.680 &   0.928 &   0.976 &   0.5 &   1.0 &   3.0 &   2.0 & 250\\ \hline
  0.720 &   0.920 &   0.952 &   1.0 &   1.0 &   3.0 &   2.0 & 250\\ \hline
  0.752 &   0.932 &   0.948 &   3.0 &   1.0 &   3.0 &   2.0 & 250\\ \hline \hline
  0.752 &   0.936 &   0.976 &   0.5 &   3.0 &   3.0 &   2.0 & 250\\ \hline
  0.828 &   0.956 &   0.964 &   1.0 &   3.0 &   3.0 &   2.0 & 250\\ \hline
  0.768 &   0.928 &   0.940 &   3.0 &   3.0 &   3.0 &   2.0 & 250\\ \hline \hline
  0.804 &   0.936 &   0.968 &   0.5 &   0.5 &   0.5 &   3.0 & 250\\ \hline
  0.764 &   0.928 &   0.964 &   1.0 &   0.5 &   0.5 &   3.0 & 250\\ \hline
  0.720 &   0.884 &   0.920 &   3.0 &   0.5 &   0.5 &   3.0 & 250\\ \hline \hline
  0.808 &   0.916 &   0.948 &   0.5 &   1.0 &   0.5 &   3.0 & 250\\ \hline
  0.819 &   0.924 &   0.948 &   1.0 &   1.0 &   0.5 &   3.0 & 249\\ \hline
  0.808 &   0.908 &   0.960 &   3.0 &   1.0 &   0.5 &   3.0 & 250\\ \hline \hline
  0.784 &   0.884 &   0.928 &   0.5 &   3.0 &   0.5 &   3.0 & 250\\ \hline
  0.859 &   0.924 &   0.944 &   1.0 &   3.0 &   0.5 &   3.0 & 249\\ \hline
  0.836 &   0.924 &   0.952 &   3.0 &   3.0 &   0.5 &   3.0 & 250\\ \hline \hline
  0.732 &   0.908 &   0.944 &   0.5 &   0.5 &   1.0 &   3.0 & 250\\ \hline
  0.672 &   0.896 &   0.940 &   1.0 &   0.5 &   1.0 &   3.0 & 250\\ \hline
  0.728 &   0.924 &   0.964 &   3.0 &   0.5 &   1.0 &   3.0 & 250\\ \hline \hline
  0.800 &   0.908 &   0.936 &   0.5 &   1.0 &   1.0 &   3.0 & 250\\ \hline
  0.768 &   0.900 &   0.912 &   1.0 &   1.0 &   1.0 &   3.0 & 250\\ \hline
  0.796 &   0.920 &   0.960 &   3.0 &   1.0 &   1.0 &   3.0 & 250\\ \hline \hline
  0.780 &   0.920 &   0.960 &   0.5 &   3.0 &   1.0 &   3.0 & 250\\ \hline
  0.784 &   0.900 &   0.936 &   1.0 &   3.0 &   1.0 &   3.0 & 250\\ \hline
  0.816 &   0.912 &   0.956 &   3.0 &   3.0 &   1.0 &   3.0 & 250\\ \hline \hline
  0.564 &   0.920 &   0.944 &   0.5 &   0.5 &   3.0 &   3.0 & 250\\ \hline
  0.628 &   0.952 &   0.944 &   1.0 &   0.5 &   3.0 &   3.0 & 250\\ \hline
  0.644 &   0.920 &   0.936 &   3.0 &   0.5 &   3.0 &   3.0 & 250\\ \hline \hline
  0.704 &   0.936 &   0.960 &   0.5 &   1.0 &   3.0 &   3.0 & 250\\ \hline
  0.680 &   0.944 &   0.952 &   1.0 &   1.0 &   3.0 &   3.0 & 250\\ \hline
  0.692 &   0.932 &   0.944 &   3.0 &   1.0 &   3.0 &   3.0 & 250\\ \hline \hline
  0.792 &   0.932 &   0.972 &   0.5 &   3.0 &   3.0 &   3.0 & 250\\ \hline
  0.776 &   0.924 &   0.956 &   1.0 &   3.0 &   3.0 &   3.0 & 250\\ \hline
  0.748 &   0.920 &   0.944 &   3.0 &   3.0 &   3.0 &   3.0 & 250\\ \hline
  \caption{Confidence Region Table for $n=100$}
  \label{confreg100}
  \end{longtable}

  \clearpage
\begin{longtable}{|r|r|r|c|c|c|c|c|}
\hline
\multicolumn{8}{|c|}{$n=250$}\\ \hline
Information & Bootstrap & Ratio & $\lambda$ & $\alpha$ & $\theta$ & $\nu$ & $N$\\ \hline
  0.900 &   0.940 &   0.964 &   0.5 &   0.5 &   0.5 &   1.0 & 250\\ \hline
  0.920 &   0.964 &   0.960 &   1.0 &   0.5 &   0.5 &   1.0 & 250\\ \hline
  0.912 &   0.936 &   0.964 &   3.0 &   0.5 &   0.5 &   1.0 & 250\\ \hline \hline
  0.884 &   0.936 &   0.960 &   0.5 &   1.0 &   0.5 &   1.0 & 250\\ \hline
  0.888 &   0.924 &   0.956 &   1.0 &   1.0 &   0.5 &   1.0 & 250\\ \hline
  0.952 &   0.968 &   0.980 &   3.0 &   1.0 &   0.5 &   1.0 & 250\\ \hline \hline
  0.920 &   0.940 &   0.972 &   0.5 &   3.0 &   0.5 &   1.0 & 249\\ \hline
  0.908 &   0.924 &   0.964 &   1.0 &   3.0 &   0.5 &   1.0 & 250\\ \hline
  0.916 &   0.948 &   0.988 &   3.0 &   3.0 &   0.5 &   1.0 & 250\\ \hline \hline
  0.860 &   0.936 &   0.952 &   0.5 &   0.5 &   1.0 &   1.0 & 250\\ \hline
  0.908 &   0.956 &   0.976 &   1.0 &   0.5 &   1.0 &   1.0 & 250\\ \hline
  0.900 &   0.940 &   0.968 &   3.0 &   0.5 &   1.0 &   1.0 & 250\\ \hline \hline
  0.908 &   0.936 &   0.972 &   0.5 &   1.0 &   1.0 &   1.0 & 250\\ \hline
  0.920 &   0.948 &   0.980 &   1.0 &   1.0 &   1.0 &   1.0 & 250\\ \hline
  0.860 &   0.920 &   0.944 &   3.0 &   1.0 &   1.0 &   1.0 & 250\\ \hline \hline
  0.936 &   0.952 &   0.960 &   0.5 &   3.0 &   1.0 &   1.0 & 250\\ \hline
  0.916 &   0.928 &   0.944 &   1.0 &   3.0 &   1.0 &   1.0 & 250\\ \hline
  0.948 &   0.956 &   0.972 &   3.0 &   3.0 &   1.0 &   1.0 & 250\\ \hline \hline
  0.856 &   0.944 &   0.976 &   0.5 &   0.5 &   3.0 &   1.0 & 250\\ \hline
  0.840 &   0.948 &   0.972 &   1.0 &   0.5 &   3.0 &   1.0 & 250\\ \hline
  0.828 &   0.964 &   0.980 &   3.0 &   0.5 &   3.0 &   1.0 & 250\\ \hline \hline
  0.904 &   0.952 &   0.980 &   0.5 &   1.0 &   3.0 &   1.0 & 250\\ \hline
  0.872 &   0.940 &   0.972 &   1.0 &   1.0 &   3.0 &   1.0 & 250\\ \hline
  0.892 &   0.948 &   0.968 &   3.0 &   1.0 &   3.0 &   1.0 & 250\\ \hline \hline
  0.888 &   0.952 &   0.968 &   0.5 &   3.0 &   3.0 &   1.0 & 250\\ \hline
  0.920 &   0.964 &   0.976 &   1.0 &   3.0 &   3.0 &   1.0 & 250\\ \hline
  0.860 &   0.928 &   0.968 &   3.0 &   3.0 &   3.0 &   1.0 & 250\\ \hline \hline
  0.848 &   0.932 &   0.964 &   0.5 &   0.5 &   0.5 &   2.0 & 250\\ \hline
  0.836 &   0.936 &   0.948 &   1.0 &   0.5 &   0.5 &   2.0 & 250\\ \hline
  0.876 &   0.932 &   0.940 &   3.0 &   0.5 &   0.5 &   2.0 & 250\\ \hline \hline
  0.876 &   0.920 &   0.964 &   0.5 &   1.0 &   0.5 &   2.0 & 250\\ \hline
  0.888 &   0.924 &   0.944 &   1.0 &   1.0 &   0.5 &   2.0 & 250\\ \hline
  0.836 &   0.932 &   0.936 &   3.0 &   1.0 &   0.5 &   2.0 & 250\\ \hline \hline
  0.896 &   0.936 &   0.964 &   0.5 &   3.0 &   0.5 &   2.0 & 250\\ \hline
  0.880 &   0.924 &   0.956 &   1.0 &   3.0 &   0.5 &   2.0 & 250\\ \hline
  0.852 &   0.916 &   0.936 &   3.0 &   3.0 &   0.5 &   2.0 & 250\\ \hline \hline
  0.848 &   0.956 &   0.952 &   0.5 &   0.5 &   1.0 &   2.0 & 250\\ \hline
  0.836 &   0.924 &   0.968 &   1.0 &   0.5 &   1.0 &   2.0 & 250\\ \hline
  0.848 &   0.936 &   0.968 &   3.0 &   0.5 &   1.0 &   2.0 & 250\\ \hline \hline
  0.900 &   0.932 &   0.960 &   0.5 &   1.0 &   1.0 &   2.0 & 250\\ \hline
  0.860 &   0.908 &   0.944 &   1.0 &   1.0 &   1.0 &   2.0 & 250\\ \hline
  0.832 &   0.892 &   0.924 &   3.0 &   1.0 &   1.0 &   2.0 & 250\\ \hline \hline
  0.876 &   0.940 &   0.952 &   0.5 &   3.0 &   1.0 &   2.0 & 250\\ \hline
  0.880 &   0.924 &   0.924 &   1.0 &   3.0 &   1.0 &   2.0 & 250\\ \hline
  0.896 &   0.968 &   0.960 &   3.0 &   3.0 &   1.0 &   2.0 & 250\\ \hline \hline
  0.696 &   0.916 &   0.932 &   0.5 &   0.5 &   3.0 &   2.0 & 250\\ \hline
  0.732 &   0.916 &   0.924 &   1.0 &   0.5 &   3.0 &   2.0 & 250\\ \hline
  0.736 &   0.928 &   0.948 &   3.0 &   0.5 &   3.0 &   2.0 & 250\\ \hline \hline
  0.820 &   0.936 &   0.960 &   0.5 &   1.0 &   3.0 &   2.0 & 250\\ \hline
  0.816 &   0.924 &   0.952 &   1.0 &   1.0 &   3.0 &   2.0 & 250\\ \hline
  0.788 &   0.932 &   0.944 &   3.0 &   1.0 &   3.0 &   2.0 & 250\\ \hline \hline
  0.876 &   0.936 &   0.960 &   0.5 &   3.0 &   3.0 &   2.0 & 250\\ \hline
  0.860 &   0.948 &   0.964 &   1.0 &   3.0 &   3.0 &   2.0 & 250\\ \hline
  0.860 &   0.928 &   0.960 &   3.0 &   3.0 &   3.0 &   2.0 & 250\\ \hline \hline
  0.864 &   0.940 &   0.972 &   0.5 &   0.5 &   0.5 &   3.0 & 250\\ \hline
  0.820 &   0.908 &   0.928 &   1.0 &   0.5 &   0.5 &   3.0 & 250\\ \hline
  0.840 &   0.944 &   0.964 &   3.0 &   0.5 &   0.5 &   3.0 & 250\\ \hline \hline
  0.884 &   0.948 &   0.960 &   0.5 &   1.0 &   0.5 &   3.0 & 250\\ \hline
  0.896 &   0.940 &   0.960 &   1.0 &   1.0 &   0.5 &   3.0 & 250\\ \hline
  0.872 &   0.936 &   0.936 &   3.0 &   1.0 &   0.5 &   3.0 & 250\\ \hline \hline
  0.912 &   0.952 &   0.952 &   0.5 &   3.0 &   0.5 &   3.0 & 250\\ \hline
  0.888 &   0.944 &   0.956 &   1.0 &   3.0 &   0.5 &   3.0 & 250\\ \hline
  0.892 &   0.896 &   0.944 &   3.0 &   3.0 &   0.5 &   3.0 & 250\\ \hline \hline
  0.804 &   0.916 &   0.964 &   0.5 &   0.5 &   1.0 &   3.0 & 250\\ \hline
  0.800 &   0.916 &   0.956 &   1.0 &   0.5 &   1.0 &   3.0 & 250\\ \hline
  0.796 &   0.936 &   0.936 &   3.0 &   0.5 &   1.0 &   3.0 & 250\\ \hline \hline
  0.916 &   0.948 &   0.956 &   0.5 &   1.0 &   1.0 &   3.0 & 250\\ \hline
  0.896 &   0.956 &   0.976 &   1.0 &   1.0 &   1.0 &   3.0 & 250\\ \hline
  0.840 &   0.916 &   0.952 &   3.0 &   1.0 &   1.0 &   3.0 & 250\\ \hline \hline
  0.888 &   0.928 &   0.932 &   0.5 &   3.0 &   1.0 &   3.0 & 250\\ \hline
  0.884 &   0.924 &   0.956 &   1.0 &   3.0 &   1.0 &   3.0 & 250\\ \hline
  0.880 &   0.920 &   0.940 &   3.0 &   3.0 &   1.0 &   3.0 & 250\\ \hline \hline
  0.728 &   0.936 &   0.928 &   0.5 &   0.5 &   3.0 &   3.0 & 250\\ \hline
  0.732 &   0.944 &   0.964 &   1.0 &   0.5 &   3.0 &   3.0 & 250\\ \hline
  0.772 &   0.940 &   0.952 &   3.0 &   0.5 &   3.0 &   3.0 & 250\\ \hline \hline
  0.820 &   0.916 &   0.944 &   0.5 &   1.0 &   3.0 &   3.0 & 250\\ \hline
  0.800 &   0.912 &   0.944 &   1.0 &   1.0 &   3.0 &   3.0 & 250\\ \hline
  0.788 &   0.908 &   0.952 &   3.0 &   1.0 &   3.0 &   3.0 & 250\\ \hline \hline
  0.804 &   0.904 &   0.940 &   0.5 &   3.0 &   3.0 &   3.0 & 250\\ \hline
  0.800 &   0.892 &   0.928 &   1.0 &   3.0 &   3.0 &   3.0 & 250\\ \hline
  0.832 &   0.928 &   0.944 &   3.0 &   3.0 &   3.0 &   3.0 & 250\\ \hline
  \caption{Confidence Region Table for $n=250$}
  \label{confreg250}
  \end{longtable}
\clearpage
\section{Z Confidence Interval and Bootstrap Confidence Interval tables}
The following tables give the average lengths and estimated coverage probabilities for the bootstrap z-interval and the bootstrap $\text{BC}_a$ interval. Each collection of parameters are separated by a double horizontal slash. The first column provides the parameter in question. The second and third column provide the average length  and estimated coverage probability of the bootstrap z-interval respectively. The fourth and fifth column provide the average length and estimated coverage of the bootstrap $\text{BC}_a$ interval respectively. In some cases, not all $\text{BC}_a$ intervals could be used. The fourth column provides the number of $\text{BC}_a$ intervals that were used to in the average length and to compute coverage probabilities (with a maximum of 250). The final column provides the value of the parameter in question that the sample was simulated from. 
% latex table generated in R 3.1.1 by xtable 1.7-4 package
% Tue Oct 07 13:09:28 2014
% latex table generated in R 3.1.1 by xtable 1.7-4 package
% Tue Oct 07 13:19:37 2014
\clearpage
\begin{longtable}{|c|r|r|r|r|c|c|}
  \hline
    \multicolumn{7}{|c|}{$n=50$}\\ \hline
Par & ZCI Len. & ZCI Cov. & BCACI Len. & BCACI Cov. & BCACI Used & True \\ 
  \hline
  $\lambda$ & 35.182 & 0.940 & 30.303 & 0.924 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.820 & 0.916 & 0.760 & 0.924 & 250 & 0.5 \\ \hline
  $\theta$ & 1.022 & 0.920 & 1.102 & 0.928 & 250 & 0.5 \\ \hline
  $\nu$ & 0.188 & 0.996 & 0.312 & 0.524 & 138 & 1.0 \\ \hline\hline
  $\lambda$ & 69.225 & 0.976 & 57.207 & 0.956 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.776 & 0.952 & 0.705 & 0.956 & 250 & 0.5 \\ \hline
  $\theta$ & 1.079 & 0.956 & 1.172 & 0.960 & 250 & 0.5 \\ \hline
  $\nu$ & 0.186 & 0.996 & 0.305 & 0.560 & 143 & 1.0 \\ \hline\hline
  $\lambda$ & 81.909 & 0.948 & 76.788 & 0.936 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.770 & 0.936 & 0.709 & 0.936 & 250 & 0.5 \\ \hline
  $\theta$ & 0.954 & 0.928 & 1.028 & 0.956 & 250 & 0.5 \\ \hline
  $\nu$ & 0.187 & 0.996 & 0.287 & 0.588 & 153 & 1.0 \\ \hline\hline
  $\lambda$ & 2.109 & 0.944 & 2.205 & 0.960 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.615 & 0.960 & 1.514 & 0.960 & 250 & 1.0 \\ \hline
  $\theta$ & 1.115 & 0.912 & 1.191 & 0.964 & 250 & 0.5 \\ \hline
  $\nu$ & 0.182 & 1.000 & 0.307 & 0.556 & 141 & 1.0 \\ \hline\hline
  $\lambda$ & 4.635 & 0.936 & 4.653 & 0.944 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.605 & 0.944 & 1.495 & 0.944 & 250 & 1.0 \\ \hline
  $\theta$ & 1.127 & 0.912 & 1.189 & 0.964 & 250 & 0.5 \\ \hline
  $\nu$ & 0.182 & 0.992 & 0.301 & 0.568 & 145 & 1.0 \\ \hline\hline
  $\lambda$ & 12.681 & 0.960 & 13.117 & 0.968 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.592 & 0.968 & 1.458 & 0.968 & 250 & 1.0 \\ \hline
  $\theta$ & 1.057 & 0.932 & 1.151 & 0.972 & 250 & 0.5 \\ \hline
  $\nu$ & 0.189 & 0.996 & 0.305 & 0.556 & 142 & 1.0 \\ \hline\hline
  $\lambda$ & 0.361 & 0.936 & 0.358 & 0.968 & 250 & 0.5 \\ \hline
  $\alpha$ & 4.775 & 0.944 & 4.549 & 0.968 & 250 & 3.0 \\ \hline
  $\theta$ & 1.145 & 0.948 & 1.248 & 0.948 & 250 & 0.5 \\ \hline
  $\nu$ & 0.187 & 0.996 & 0.321 & 0.504 & 132 & 1.0 \\ \hline\hline
  $\lambda$ & 0.758 & 0.932 & 0.753 & 0.960 & 250 & 1.0 \\ \hline
  $\alpha$ & 4.682 & 0.948 & 4.459 & 0.960 & 250 & 3.0 \\ \hline
  $\theta$ & 1.195 & 0.912 & 1.285 & 0.956 & 250 & 0.5 \\ \hline
  $\nu$ & 0.187 & 0.992 & 0.319 & 0.508 & 130 & 1.0 \\ \hline\hline
  $\lambda$ & 2.121 & 0.932 & 2.024 & 0.952 & 250 & 3.0 \\ \hline
  $\alpha$ & 4.918 & 0.948 & 4.660 & 0.952 & 250 & 3.0 \\ \hline
  $\theta$ & 1.100 & 0.928 & 1.174 & 0.964 & 250 & 0.5 \\ \hline
  $\nu$ & 0.179 & 1.000 & 0.307 & 0.492 & 128 & 1.0 \\ \hline\hline
  $\lambda$ & 21.030 & 0.956 & 20.386 & 0.944 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.745 & 0.968 & 0.702 & 0.944 & 250 & 0.5 \\ \hline
  $\theta$ & 2.527 & 0.936 & 2.619 & 0.956 & 250 & 1.0 \\ \hline
  $\nu$ & 0.187 & 1.000 & 0.307 & 0.572 & 149 & 1.0 \\ \hline\hline
  $\lambda$ & 34.065 & 0.960 & 32.652 & 0.968 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.715 & 0.980 & 0.673 & 0.968 & 250 & 0.5 \\ \hline
  $\theta$ & 2.529 & 0.956 & 2.665 & 0.968 & 250 & 1.0 \\ \hline
  $\nu$ & 0.181 & 1.000 & 0.300 & 0.552 & 142 & 1.0 \\ \hline\hline
  $\lambda$ & 88.971 & 0.976 & 70.812 & 0.956 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.671 & 0.976 & 0.630 & 0.956 & 250 & 0.5 \\ \hline
  $\theta$ & 2.425 & 0.944 & 2.464 & 0.968 & 250 & 1.0 \\ \hline
  $\nu$ & 0.171 & 1.000 & 0.288 & 0.540 & 135 & 1.0 \\ \hline\hline
  $\lambda$ & 2.693 & 0.952 & 2.786 & 0.968 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.516 & 0.964 & 1.435 & 0.968 & 250 & 1.0 \\ \hline
  $\theta$ & 2.844 & 0.964 & 2.843 & 0.960 & 250 & 1.0 \\ \hline
  $\nu$ & 0.182 & 0.996 & 0.306 & 0.536 & 141 & 1.0 \\ \hline\hline
  $\lambda$ & 6.558 & 0.948 & 6.635 & 0.936 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.501 & 0.956 & 1.419 & 0.936 & 250 & 1.0 \\ \hline
  $\theta$ & 3.047 & 0.956 & 3.173 & 0.956 & 250 & 1.0 \\ \hline
  $\nu$ & 0.198 & 0.996 & 0.331 & 0.548 & 145 & 1.0 \\ \hline\hline
  $\lambda$ & 13.671 & 0.944 & 14.815 & 0.940 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.520 & 0.968 & 1.430 & 0.940 & 250 & 1.0 \\ \hline
  $\theta$ & 2.686 & 0.932 & 2.797 & 0.960 & 250 & 1.0 \\ \hline
  $\nu$ & 0.184 & 0.992 & 0.317 & 0.524 & 137 & 1.0 \\ \hline\hline
  $\lambda$ & 0.481 & 0.944 & 0.488 & 0.948 & 250 & 0.5 \\ \hline
  $\alpha$ & 4.386 & 0.956 & 4.034 & 0.948 & 250 & 3.0 \\ \hline
  $\theta$ & 4.281 & 0.952 & 5.116 & 0.932 & 250 & 1.0 \\ \hline
  $\nu$ & 0.195 & 1.000 & 0.322 & 0.560 & 146 & 1.0 \\ \hline\hline
  $\lambda$ & 0.775 & 0.948 & 0.768 & 0.948 & 250 & 1.0 \\ \hline
  $\alpha$ & 4.578 & 0.976 & 4.317 & 0.948 & 250 & 3.0 \\ \hline
  $\theta$ & 2.947 & 0.920 & 3.188 & 0.968 & 250 & 1.0 \\ \hline
  $\nu$ & 0.180 & 0.996 & 0.313 & 0.480 & 123 & 1.0 \\ \hline\hline
  $\lambda$ & 2.705 & 0.960 & 2.474 & 0.960 & 250 & 3.0 \\ \hline
  $\alpha$ & 4.343 & 0.972 & 4.139 & 0.960 & 250 & 3.0 \\ \hline
  $\theta$ & 3.509 & 0.956 & 3.430 & 0.964 & 250 & 1.0 \\ \hline
  $\nu$ & 0.183 & 1.000 & 0.314 & 0.504 & 130 & 1.0 \\ \hline\hline
  $\lambda$ & 16.087 & 0.964 & 16.460 & 0.936 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.590 & 0.964 & 0.596 & 0.936 & 250 & 0.5 \\ \hline
  $\theta$ & 10.690 & 0.948 & 10.828 & 0.936 & 250 & 3.0 \\ \hline
  $\nu$ & 0.186 & 0.996 & 0.296 & 0.568 & 147 & 1.0 \\ \hline\hline
  $\lambda$ & 23.130 & 0.964 & 25.713 & 0.944 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.580 & 0.968 & 0.594 & 0.944 & 250 & 0.5 \\ \hline
  $\theta$ & 10.408 & 0.960 & 10.567 & 0.956 & 250 & 3.0 \\ \hline
  $\nu$ & 0.177 & 1.000 & 0.292 & 0.544 & 140 & 1.0 \\ \hline\hline
  $\lambda$ & 54.289 & 0.960 & 54.163 & 0.944 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.586 & 0.976 & 0.601 & 0.944 & 250 & 0.5 \\ \hline
  $\theta$ & 9.508 & 0.960 & 10.203 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 0.180 & 0.992 & 0.307 & 0.496 & 127 & 1.0 \\ \hline\hline
  $\lambda$ & 3.067 & 0.968 & 3.479 & 0.956 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.158 & 0.968 & 1.138 & 0.956 & 250 & 1.0 \\ \hline
  $\theta$ & 15.619 & 0.968 & 17.117 & 0.972 & 250 & 3.0 \\ \hline
  $\nu$ & 0.185 & 1.000 & 0.299 & 0.540 & 139 & 1.0 \\ \hline\hline
  $\lambda$ & 5.575 & 0.964 & 6.260 & 0.960 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.195 & 0.976 & 1.195 & 0.960 & 250 & 1.0 \\ \hline
  $\theta$ & 15.673 & 0.976 & 16.192 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 0.188 & 1.000 & 0.312 & 0.520 & 132 & 1.0 \\ \hline\hline
  $\lambda$ & 16.148 & 0.976 & 18.526 & 0.964 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.172 & 0.980 & 1.204 & 0.964 & 250 & 1.0 \\ \hline
  $\theta$ & 14.161 & 0.980 & 14.999 & 0.976 & 250 & 3.0 \\ \hline
  $\nu$ & 0.191 & 0.988 & 0.331 & 0.484 & 129 & 1.0 \\ \hline\hline
  $\lambda$ & 0.557 & 0.956 & 0.568 & 0.924 & 250 & 0.5 \\ \hline
  $\alpha$ & 3.681 & 0.956 & 3.596 & 0.924 & 250 & 3.0 \\ \hline
  $\theta$ & 24.635 & 0.964 & 29.870 & 0.944 & 250 & 3.0 \\ \hline
  $\nu$ & 0.179 & 1.000 & 0.307 & 0.504 & 130 & 1.0 \\ \hline\hline
  $\lambda$ & 1.105 & 0.936 & 1.120 & 0.928 & 250 & 1.0 \\ \hline
  $\alpha$ & 3.729 & 0.964 & 3.664 & 0.928 & 250 & 3.0 \\ \hline
  $\theta$ & 23.240 & 0.956 & 28.096 & 0.952 & 250 & 3.0 \\ \hline
  $\nu$ & 0.190 & 0.992 & 0.316 & 0.548 & 141 & 1.0 \\ \hline\hline
  $\lambda$ & 3.351 & 0.940 & 3.239 & 0.948 & 250 & 3.0 \\ \hline
  $\alpha$ & 3.607 & 0.960 & 3.619 & 0.948 & 250 & 3.0 \\ \hline
  $\theta$ & 24.837 & 0.948 & 26.172 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 0.185 & 1.000 & 0.317 & 0.512 & 135 & 1.0 \\ \hline\hline
  $\lambda$ & 45.104 & 0.972 & 68.786 & 0.968 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.901 & 0.968 & 0.790 & 0.968 & 250 & 0.5 \\ \hline
  $\theta$ & 1.201 & 0.944 & 1.333 & 0.956 & 250 & 0.5 \\ \hline
  $\nu$ & 1.166 & 0.932 & 1.205 & 0.936 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 90.661 & 0.968 & 126.738 & 0.940 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.922 & 0.952 & 0.815 & 0.940 & 250 & 0.5 \\ \hline
  $\theta$ & 1.143 & 0.916 & 1.259 & 0.952 & 250 & 0.5 \\ \hline
  $\nu$ & 1.143 & 0.940 & 1.179 & 0.964 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 269.719 & 0.956 & 421.338 & 0.936 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.887 & 0.936 & 0.780 & 0.936 & 250 & 0.5 \\ \hline
  $\theta$ & 1.236 & 0.936 & 1.351 & 0.932 & 250 & 0.5 \\ \hline
  $\nu$ & 1.121 & 0.928 & 1.154 & 0.948 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 3.326 & 0.944 & 5.030 & 0.952 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.774 & 0.956 & 1.562 & 0.952 & 250 & 1.0 \\ \hline
  $\theta$ & 1.230 & 0.924 & 1.381 & 0.956 & 250 & 0.5 \\ \hline
  $\nu$ & 1.146 & 0.936 & 1.184 & 0.952 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 7.055 & 0.948 & 9.870 & 0.920 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.872 & 0.940 & 1.653 & 0.920 & 250 & 1.0 \\ \hline
  $\theta$ & 1.143 & 0.912 & 1.268 & 0.944 & 250 & 0.5 \\ \hline
  $\nu$ & 1.161 & 0.948 & 1.194 & 0.936 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 20.016 & 0.944 & 25.209 & 0.940 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.843 & 0.944 & 1.629 & 0.940 & 250 & 1.0 \\ \hline
  $\theta$ & 1.158 & 0.904 & 1.293 & 0.952 & 250 & 0.5 \\ \hline
  $\nu$ & 1.160 & 0.932 & 1.198 & 0.936 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.375 & 0.956 & 0.376 & 0.984 & 250 & 0.5 \\ \hline
  $\alpha$ & 5.480 & 0.956 & 4.865 & 0.984 & 250 & 3.0 \\ \hline
  $\theta$ & 1.154 & 0.932 & 1.277 & 0.972 & 250 & 0.5 \\ \hline
  $\nu$ & 1.166 & 0.960 & 1.204 & 0.948 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.853 & 0.964 & 0.870 & 0.960 & 250 & 1.0 \\ \hline
  $\alpha$ & 5.528 & 0.940 & 4.874 & 0.960 & 250 & 3.0 \\ \hline
  $\theta$ & 1.282 & 0.932 & 1.451 & 0.940 & 250 & 0.5 \\ \hline
  $\nu$ & 1.118 & 0.932 & 1.155 & 0.960 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 2.475 & 0.952 & 2.431 & 0.976 & 250 & 3.0 \\ \hline
  $\alpha$ & 5.414 & 0.972 & 4.755 & 0.976 & 250 & 3.0 \\ \hline
  $\theta$ & 1.247 & 0.928 & 1.381 & 0.944 & 250 & 0.5 \\ \hline
  $\nu$ & 1.158 & 0.940 & 1.196 & 0.956 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 49.933 & 0.964 & 78.497 & 0.948 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.848 & 0.964 & 0.738 & 0.948 & 250 & 0.5 \\ \hline
  $\theta$ & 3.177 & 0.960 & 3.710 & 0.948 & 250 & 1.0 \\ \hline
  $\nu$ & 1.137 & 0.944 & 1.170 & 0.956 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 98.986 & 0.948 & 135.780 & 0.952 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.870 & 0.960 & 0.752 & 0.952 & 250 & 0.5 \\ \hline
  $\theta$ & 3.018 & 0.924 & 3.518 & 0.932 & 250 & 1.0 \\ \hline
  $\nu$ & 1.133 & 0.928 & 1.168 & 0.936 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 162.169 & 0.956 & 244.852 & 0.956 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.838 & 0.988 & 0.733 & 0.956 & 250 & 0.5 \\ \hline
  $\theta$ & 2.956 & 0.948 & 3.430 & 0.944 & 250 & 1.0 \\ \hline
  $\nu$ & 1.123 & 0.948 & 1.157 & 0.956 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 4.842 & 0.944 & 7.589 & 0.916 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.697 & 0.956 & 1.491 & 0.916 & 250 & 1.0 \\ \hline
  $\theta$ & 3.540 & 0.920 & 4.341 & 0.932 & 250 & 1.0 \\ \hline
  $\nu$ & 1.133 & 0.956 & 1.172 & 0.964 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 8.944 & 0.964 & 13.009 & 0.952 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.688 & 0.976 & 1.465 & 0.952 & 250 & 1.0 \\ \hline
  $\theta$ & 3.552 & 0.968 & 4.319 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 1.127 & 0.932 & 1.163 & 0.956 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 30.134 & 0.968 & 42.602 & 0.940 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.692 & 0.960 & 1.465 & 0.940 & 250 & 1.0 \\ \hline
  $\theta$ & 3.612 & 0.956 & 4.371 & 0.936 & 250 & 1.0 \\ \hline
  $\nu$ & 1.125 & 0.928 & 1.161 & 0.936 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.504 & 0.952 & 0.493 & 0.956 & 250 & 0.5 \\ \hline
  $\alpha$ & 4.997 & 0.980 & 4.354 & 0.956 & 250 & 3.0 \\ \hline
  $\theta$ & 4.079 & 0.948 & 5.400 & 0.936 & 250 & 1.0 \\ \hline
  $\nu$ & 1.134 & 0.920 & 1.172 & 0.944 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.049 & 0.920 & 1.483 & 0.924 & 250 & 1.0 \\ \hline
  $\alpha$ & 5.159 & 0.956 & 4.521 & 0.924 & 250 & 3.0 \\ \hline
  $\theta$ & 4.174 & 0.928 & 5.755 & 0.916 & 250 & 1.0 \\ \hline
  $\nu$ & 1.139 & 0.956 & 1.174 & 0.960 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 2.977 & 0.944 & 2.953 & 0.964 & 250 & 3.0 \\ \hline
  $\alpha$ & 5.055 & 0.972 & 4.422 & 0.964 & 250 & 3.0 \\ \hline
  $\theta$ & 4.151 & 0.948 & 5.078 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 1.134 & 0.932 & 1.171 & 0.944 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 41.140 & 0.972 & 60.391 & 0.940 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.715 & 0.956 & 0.658 & 0.940 & 250 & 0.5 \\ \hline
  $\theta$ & 16.949 & 0.952 & 21.008 & 0.924 & 250 & 3.0 \\ \hline
  $\nu$ & 1.112 & 0.912 & 1.152 & 0.932 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 58.121 & 0.984 & 89.772 & 0.948 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.707 & 0.972 & 0.654 & 0.948 & 250 & 0.5 \\ \hline
  $\theta$ & 15.338 & 0.964 & 19.471 & 0.952 & 250 & 3.0 \\ \hline
  $\nu$ & 1.140 & 0.924 & 1.178 & 0.948 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 152.526 & 0.980 & 204.303 & 0.952 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.697 & 0.948 & 0.640 & 0.952 & 250 & 0.5 \\ \hline
  $\theta$ & 15.205 & 0.972 & 19.122 & 0.936 & 250 & 3.0 \\ \hline
  $\nu$ & 1.112 & 0.936 & 1.149 & 0.932 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 4.460 & 0.968 & 6.696 & 0.956 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.426 & 0.984 & 1.330 & 0.956 & 250 & 1.0 \\ \hline
  $\theta$ & 22.748 & 0.960 & 31.112 & 0.980 & 250 & 3.0 \\ \hline
  $\nu$ & 1.140 & 0.940 & 1.177 & 0.964 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 9.015 & 0.936 & 13.449 & 0.928 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.525 & 0.976 & 1.402 & 0.928 & 250 & 1.0 \\ \hline
  $\theta$ & 21.769 & 0.940 & 30.004 & 0.920 & 250 & 3.0 \\ \hline
  $\nu$ & 1.138 & 0.940 & 1.181 & 0.948 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 28.874 & 0.964 & 42.060 & 0.952 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.434 & 0.968 & 1.326 & 0.952 & 250 & 1.0 \\ \hline
  $\theta$ & 24.487 & 0.984 & 33.065 & 0.948 & 250 & 3.0 \\ \hline
  $\nu$ & 1.124 & 0.904 & 1.161 & 0.948 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.689 & 0.920 & 0.724 & 0.936 & 250 & 0.5 \\ \hline
  $\alpha$ & 4.629 & 0.976 & 4.284 & 0.936 & 250 & 3.0 \\ \hline
  $\theta$ & 39.431 & 0.948 & 56.595 & 0.952 & 250 & 3.0 \\ \hline
  $\nu$ & 1.126 & 0.916 & 1.163 & 0.932 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.656 & 0.960 & 1.771 & 0.932 & 250 & 1.0 \\ \hline
  $\alpha$ & 4.319 & 0.956 & 4.015 & 0.932 & 250 & 3.0 \\ \hline
  $\theta$ & 55.329 & 0.972 & 77.540 & 0.928 & 250 & 3.0 \\ \hline
  $\nu$ & 1.157 & 0.948 & 1.194 & 0.928 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 4.492 & 0.948 & 4.804 & 0.940 & 250 & 3.0 \\ \hline
  $\alpha$ & 4.370 & 0.972 & 4.074 & 0.940 & 250 & 3.0 \\ \hline
  $\theta$ & 49.043 & 0.972 & 69.986 & 0.916 & 250 & 3.0 \\ \hline
  $\nu$ & 1.137 & 0.948 & 1.174 & 0.960 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 103.793 & 0.984 & 186.577 & 0.952 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.897 & 0.952 & 0.796 & 0.952 & 250 & 0.5 \\ \hline
  $\theta$ & 1.319 & 0.944 & 1.446 & 0.936 & 250 & 0.5 \\ \hline
  $\nu$ & 1.849 & 0.952 & 1.903 & 0.952 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 104.067 & 0.980 & 183.138 & 0.932 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.927 & 0.944 & 0.825 & 0.932 & 250 & 0.5 \\ \hline
  $\theta$ & 1.210 & 0.936 & 1.315 & 0.948 & 250 & 0.5 \\ \hline
  $\nu$ & 1.829 & 0.940 & 1.879 & 0.948 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 307.246 & 0.952 & 317.437 & 0.932 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.929 & 0.932 & 0.835 & 0.932 & 250 & 0.5 \\ \hline
  $\theta$ & 1.238 & 0.924 & 1.340 & 0.932 & 250 & 0.5 \\ \hline
  $\nu$ & 1.867 & 0.916 & 1.927 & 0.916 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 4.153 & 0.956 & 5.810 & 0.944 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.840 & 0.948 & 1.636 & 0.944 & 250 & 1.0 \\ \hline
  $\theta$ & 1.292 & 0.936 & 1.430 & 0.928 & 250 & 0.5 \\ \hline
  $\nu$ & 1.809 & 0.876 & 1.864 & 0.920 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 6.209 & 0.968 & 8.407 & 0.956 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.854 & 0.964 & 1.653 & 0.956 & 250 & 1.0 \\ \hline
  $\theta$ & 1.233 & 0.924 & 1.353 & 0.964 & 250 & 0.5 \\ \hline
  $\nu$ & 1.837 & 0.968 & 1.894 & 0.984 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 24.659 & 0.972 & 32.413 & 0.952 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.769 & 0.956 & 1.578 & 0.952 & 250 & 1.0 \\ \hline
  $\theta$ & 1.391 & 0.948 & 1.543 & 0.936 & 250 & 0.5 \\ \hline
  $\nu$ & 1.849 & 0.948 & 1.907 & 0.952 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.489 & 0.956 & 0.501 & 0.956 & 250 & 0.5 \\ \hline
  $\alpha$ & 5.417 & 0.952 & 4.819 & 0.956 & 250 & 3.0 \\ \hline
  $\theta$ & 1.467 & 0.936 & 1.620 & 0.944 & 250 & 0.5 \\ \hline
  $\nu$ & 1.822 & 0.964 & 1.880 & 0.960 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.871 & 0.968 & 0.849 & 0.964 & 250 & 1.0 \\ \hline
  $\alpha$ & 5.461 & 0.960 & 4.857 & 0.964 & 250 & 3.0 \\ \hline
  $\theta$ & 1.321 & 0.936 & 1.454 & 0.948 & 250 & 0.5 \\ \hline
  $\nu$ & 1.852 & 0.944 & 1.899 & 0.976 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 2.968 & 0.952 & 2.847 & 0.944 & 250 & 3.0 \\ \hline
  $\alpha$ & 5.290 & 0.948 & 4.728 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 1.516 & 0.920 & 1.680 & 0.932 & 250 & 0.5 \\ \hline
  $\nu$ & 1.859 & 0.944 & 1.911 & 0.944 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 48.889 & 0.980 & 82.648 & 0.972 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.831 & 0.980 & 0.729 & 0.972 & 250 & 0.5 \\ \hline
  $\theta$ & 3.150 & 0.960 & 3.644 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 1.893 & 0.948 & 1.946 & 0.956 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 99.803 & 0.972 & 152.826 & 0.944 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.836 & 0.972 & 0.734 & 0.944 & 250 & 0.5 \\ \hline
  $\theta$ & 3.232 & 0.968 & 3.706 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 1.837 & 0.916 & 1.892 & 0.940 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 261.670 & 0.964 & 346.254 & 0.948 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.818 & 0.960 & 0.722 & 0.948 & 250 & 0.5 \\ \hline
  $\theta$ & 3.226 & 0.960 & 3.698 & 0.948 & 250 & 1.0 \\ \hline
  $\nu$ & 1.853 & 0.964 & 1.907 & 0.976 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 7.398 & 0.952 & 10.755 & 0.936 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.693 & 0.968 & 1.475 & 0.936 & 250 & 1.0 \\ \hline
  $\theta$ & 3.872 & 0.936 & 4.498 & 0.944 & 250 & 1.0 \\ \hline
  $\nu$ & 1.844 & 0.932 & 1.897 & 0.956 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 6.452 & 0.952 & 9.292 & 0.956 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.734 & 0.988 & 1.513 & 0.956 & 250 & 1.0 \\ \hline
  $\theta$ & 3.239 & 0.940 & 3.831 & 0.968 & 250 & 1.0 \\ \hline
  $\nu$ & 1.864 & 0.936 & 1.921 & 0.932 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 20.865 & 0.952 & 28.742 & 0.964 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.714 & 0.972 & 1.491 & 0.964 & 250 & 1.0 \\ \hline
  $\theta$ & 3.256 & 0.952 & 3.814 & 0.936 & 250 & 1.0 \\ \hline
  $\nu$ & 1.889 & 0.944 & 1.946 & 0.960 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.490 & 0.956 & 0.489 & 0.932 & 250 & 0.5 \\ \hline
  $\alpha$ & 5.167 & 0.956 & 4.551 & 0.932 & 250 & 3.0 \\ \hline
  $\theta$ & 4.008 & 0.956 & 4.891 & 0.928 & 250 & 1.0 \\ \hline
  $\nu$ & 1.838 & 0.944 & 1.892 & 0.972 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 1.026 & 0.952 & 1.007 & 0.936 & 250 & 1.0 \\ \hline
  $\alpha$ & 5.157 & 0.968 & 4.558 & 0.936 & 250 & 3.0 \\ \hline
  $\theta$ & 4.293 & 0.928 & 5.106 & 0.932 & 250 & 1.0 \\ \hline
  $\nu$ & 1.814 & 0.940 & 1.860 & 0.952 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 3.427 & 0.948 & 3.399 & 0.924 & 250 & 3.0 \\ \hline
  $\alpha$ & 5.143 & 0.956 & 4.527 & 0.924 & 250 & 3.0 \\ \hline
  $\theta$ & 5.039 & 0.948 & 6.286 & 0.916 & 250 & 1.0 \\ \hline
  $\nu$ & 1.835 & 0.952 & 1.893 & 0.944 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 35.670 & 0.984 & 54.559 & 0.960 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.690 & 0.980 & 0.640 & 0.960 & 250 & 0.5 \\ \hline
  $\theta$ & 15.986 & 0.984 & 19.948 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 1.796 & 0.940 & 1.851 & 0.952 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 86.706 & 0.968 & 117.324 & 0.928 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.692 & 0.956 & 0.640 & 0.928 & 250 & 0.5 \\ \hline
  $\theta$ & 18.130 & 0.964 & 22.264 & 0.932 & 250 & 3.0 \\ \hline
  $\nu$ & 1.796 & 0.908 & 1.859 & 0.920 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 157.436 & 0.972 & 221.447 & 0.932 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.715 & 0.976 & 0.657 & 0.932 & 250 & 0.5 \\ \hline
  $\theta$ & 15.317 & 0.952 & 18.896 & 0.940 & 250 & 3.0 \\ \hline
  $\nu$ & 1.841 & 0.960 & 1.898 & 0.960 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 5.992 & 0.968 & 8.620 & 0.940 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.418 & 0.952 & 1.324 & 0.940 & 250 & 1.0 \\ \hline
  $\theta$ & 26.983 & 0.972 & 34.994 & 0.936 & 250 & 3.0 \\ \hline
  $\nu$ & 1.830 & 0.912 & 1.886 & 0.928 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 10.685 & 0.964 & 15.604 & 0.948 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.420 & 0.980 & 1.323 & 0.948 & 250 & 1.0 \\ \hline
  $\theta$ & 30.423 & 0.976 & 38.800 & 0.952 & 250 & 3.0 \\ \hline
  $\nu$ & 1.811 & 0.892 & 1.867 & 0.928 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 26.452 & 0.972 & 38.557 & 0.976 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.421 & 0.988 & 1.328 & 0.976 & 250 & 1.0 \\ \hline
  $\theta$ & 23.906 & 0.964 & 33.388 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 1.822 & 0.920 & 1.888 & 0.932 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.717 & 0.968 & 0.743 & 0.968 & 250 & 0.5 \\ \hline
  $\alpha$ & 4.220 & 0.972 & 3.966 & 0.968 & 250 & 3.0 \\ \hline
  $\theta$ & 41.610 & 0.984 & 59.093 & 0.952 & 250 & 3.0 \\ \hline
  $\nu$ & 1.844 & 0.956 & 1.897 & 0.976 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 1.578 & 0.932 & 1.624 & 0.944 & 250 & 1.0 \\ \hline
  $\alpha$ & 4.189 & 0.968 & 3.922 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 49.195 & 0.960 & 66.168 & 0.956 & 250 & 3.0 \\ \hline
  $\nu$ & 1.845 & 0.948 & 1.906 & 0.956 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 4.213 & 0.952 & 4.352 & 0.964 & 250 & 3.0 \\ \hline
  $\alpha$ & 4.287 & 0.984 & 4.024 & 0.964 & 250 & 3.0 \\ \hline
  $\theta$ & 38.734 & 0.964 & 55.065 & 0.956 & 250 & 3.0 \\ \hline
  $\nu$ & 1.883 & 0.952 & 1.949 & 0.952 & 250 & 3.0 \\ \hline
  \caption{Bootstrap $z$ vs. The Bootstrap $\text{BC}_a$ Table for $n=50$}
  \label{bstab50}
\end{longtable}
\clearpage
% latex table generated in R 3.1.1 by xtable 1.7-4 package
% Tue Oct 07 13:35:35 2014
\begin{longtable}{|c|r|r|r|r|c|c|}
  \hline
      \multicolumn{7}{|c|}{$n=100$}\\ \hline
Par & ZCI Len. & ZCI Cov. & BCACI Len. & BCACI Cov. & BCACI Used & True \\ 
  \hline
$\lambda$ & 2.891 & 0.948 & 2.960 & 0.936 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.477 & 0.956 & 0.464 & 0.936 & 250 & 0.5 \\ \hline
  $\theta$ & 0.632 & 0.932 & 0.657 & 0.940 & 250 & 0.5 \\ \hline
  $\nu$ & 0.123 & 0.988 & 0.200 & 0.516 & 134 & 1.0 \\ \hline\hline
  $\lambda$ & 5.511 & 0.936 & 5.473 & 0.940 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.485 & 0.960 & 0.471 & 0.940 & 250 & 0.5 \\ \hline
  $\theta$ & 0.598 & 0.932 & 0.627 & 0.948 & 250 & 0.5 \\ \hline
  $\nu$ & 0.118 & 0.992 & 0.193 & 0.500 & 128 & 1.0 \\ \hline\hline
  $\lambda$ & 12.422 & 0.976 & 13.007 & 0.976 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.451 & 0.972 & 0.444 & 0.976 & 250 & 0.5 \\ \hline
  $\theta$ & 0.568 & 0.952 & 0.602 & 0.976 & 250 & 0.5 \\ \hline
  $\nu$ & 0.118 & 1.000& 0.174 & 0.564 & 147 & 1.0\\ \hline\hline
  $\lambda$ & 0.831 & 0.924 & 0.805 & 0.940 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.976 & 0.944 & 0.958 & 0.940 & 250 & 1.0\\ \hline
  $\theta$ & 0.647 & 0.916 & 0.669 & 0.948 & 250 & 0.5 \\ \hline
  $\nu$ & 0.126 & 0.988 & 0.199 & 0.552 & 147 & 1.000\\ \hline\hline
  $\lambda$ & 1.654 & 0.920 & 1.645 & 0.916 & 250 & 1.0\\ \hline
  $\alpha$ & 0.986 & 0.932 & 0.964 & 0.916 & 250 & 1.0\\ \hline
  $\theta$ & 0.635 & 0.912 & 0.661 & 0.928 & 250 & 0.5 \\ \hline
  $\nu$ & 0.123 & 0.992 & 0.205 & 0.484 & 124 & 1.0\\ \hline\hline
  $\lambda$ & 4.338 & 0.956 & 4.371 & 0.952 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.950 & 0.956 & 0.927 & 0.952 & 250 & 1.0 \\ \hline
  $\theta$ & 0.598 & 0.944 & 0.636 & 0.952 & 250 & 0.5 \\ \hline
  $\nu$ & 0.127 & 1.000& 0.199 & 0.580 & 150 & 1.0 \\ \hline\hline
  $\lambda$ & 0.194 & 0.932 & 0.205 & 0.956 & 250 & 0.5 \\ \hline
  $\alpha$ & 2.842 & 0.948 & 2.862 & 0.956 & 250 & 3.0 \\ \hline
  $\theta$ & 0.606 & 0.948 & 0.654 & 0.964 & 250 & 0.5 \\ \hline
  $\nu$ & 0.118 & 0.996 & 0.184 & 0.536 & 137 & 1.0 \\ \hline\hline
  $\lambda$ & 0.402 & 0.948 & 0.422 & 0.952 & 250 & 1.0 \\ \hline
  $\alpha$ & 2.805 & 0.972 & 2.792 & 0.952 & 250 & 3.0 \\ \hline
  $\theta$ & 0.635 & 0.952 & 0.682 & 0.956 & 250 & 0.5 \\ \hline
  $\nu$ & 0.117 & 0.996 & 0.192 & 0.520 & 134 & 1.0 \\ \hline\hline
  $\lambda$ & 1.243 & 0.956 & 1.282 & 0.956 & 250 & 3.0 \\ \hline
  $\alpha$ & 2.815 & 0.968 & 2.754 & 0.956 & 250 & 3.0 \\ \hline
  $\theta$ & 0.659 & 0.964 & 0.701 & 0.972 & 250 & 0.5 \\ \hline
  $\nu$ & 0.123 & 1.000& 0.198 & 0.532 & 138 & 1.0 \\ \hline\hline
  $\lambda$ & 3.569 & 0.984 & 3.685 & 0.964 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.417 & 0.972 & 0.425 & 0.964 & 250 & 0.5 \\ \hline
  $\theta$ & 1.471 & 0.980 & 1.465 & 0.976 & 250 & 1.0 \\ \hline
  $\nu$ & 0.124 & 0.996 & 0.196 & 0.560 & 143 & 1.0 \\ \hline\hline
  $\lambda$ & 7.177 & 0.952 & 6.900 & 0.964 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.412 & 0.960 & 0.419 & 0.964 & 250 & 0.5 \\ \hline
  $\theta$ & 1.449 & 0.956 & 1.444 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 0.117 & 1.000& 0.186 & 0.560 & 143 & 1.0 \\ \hline\hline
  $\lambda$ & 18.544 & 0.952 & 19.106 & 0.952 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.396 & 0.968 & 0.402 & 0.952 & 250 & 0.5 \\ \hline
  $\theta$ & 1.420 & 0.960 & 1.434 & 0.960 & 250 & 1.0 \\ \hline
  $\nu$ & 0.123 & 0.988 & 0.199 & 0.508 & 131 & 1.0 \\ \hline\hline
  $\lambda$ & 0.999 & 0.928 & 0.939 & 0.932 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.830 & 0.964 & 0.843 & 0.932 & 250 & 1.0 \\ \hline
  $\theta$ & 1.619 & 0.924 & 1.568 & 0.964 & 250 & 1.0 \\ \hline
  $\nu$ & 0.121 & 0.996 & 0.193 & 0.556 & 144 & 1.0 \\ \hline\hline
  $\lambda$ & 1.887 & 0.968 & 1.774 & 0.948 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.826 & 0.964 & 0.843 & 0.948 & 250 & 1.0 \\ \hline
  $\theta$ & 1.592 & 0.964 & 1.563 & 0.956 & 250 & 1.0 \\ \hline
  $\nu$ & 0.124 & 0.988 & 0.203 & 0.512 & 133 & 1.0 \\ \hline\hline
  $\lambda$ & 5.332 & 0.964 & 5.216 & 0.952 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.805 & 0.964 & 0.819 & 0.952 & 250 & 1.0 \\ \hline
  $\theta$ & 1.526 & 0.972 & 1.516 & 0.968 & 250 & 1.0 \\ \hline
  $\nu$ & 0.125 & 1.000& 0.201 & 0.556 & 142 & 1.0 \\ \hline\hline
  $\lambda$ & 0.209 & 0.952 & 0.220 & 0.944 & 250 & 0.5 \\ \hline
  $\alpha$ & 2.449 & 0.980 & 2.487 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 1.493 & 0.956 & 1.575 & 0.964 & 250 & 1.0 \\ \hline
  $\nu$ & 0.115 & 1.000& 0.185 & 0.512 & 131 & 1.0 \\ \hline\hline
  $\lambda$ & 0.443 & 0.960 & 0.458 & 0.956 & 250 & 1.0 \\ \hline
  $\alpha$ & 2.468 & 0.972 & 2.485 & 0.956 & 250 & 3.0 \\ \hline
  $\theta$ & 1.597 & 0.956 & 1.638 & 0.948 & 250 & 1.0 \\ \hline
  $\nu$ & 0.124 & 0.992 & 0.200 & 0.532 & 139 & 1.0 \\ \hline\hline
  $\lambda$ & 1.291 & 0.952 & 1.297 & 0.932 & 250 & 3.0 \\ \hline
  $\alpha$ & 2.538 & 0.960 & 2.585 & 0.932 & 250 & 3.0 \\ \hline
  $\theta$ & 1.574 & 0.952 & 1.553 & 0.956 & 250 & 1.0 \\ \hline
  $\nu$ & 0.119 & 0.996 & 0.193 & 0.504 & 133 & 1.0 \\ \hline\hline
  $\lambda$ & 4.055 & 0.964 & 4.229 & 0.968 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.358 & 0.976 & 0.382 & 0.968 & 250 & 0.5 \\ \hline
  $\theta$ & 6.783 & 0.968 & 6.437 & 0.968 & 250 & 3.0 \\ \hline
  $\nu$ & 0.116 & 0.996 & 0.172 & 0.560 & 145 & 1.0 \\ \hline\hline
  $\lambda$ & 7.781 & 0.964 & 8.106 & 0.972 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.362 & 0.976 & 0.387 & 0.972 & 250 & 0.5 \\ \hline
  $\theta$ & 6.702 & 0.968 & 6.290 & 0.976 & 250 & 3.0 \\ \hline
  $\nu$ & 0.118 & 0.988 & 0.179 & 0.576 & 150 & 1.0 \\ \hline\hline
  $\lambda$ & 26.456 & 0.968 & 26.280 & 0.952 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.355 & 0.960 & 0.380 & 0.952 & 250 & 0.5 \\ \hline
  $\theta$ & 6.792 & 0.972 & 6.574 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 0.118 & 0.992 & 0.190 & 0.540 & 141 & 1.0 \\ \hline\hline
  $\lambda$ & 1.104 & 0.964 & 1.043 & 0.960 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.739 & 0.968 & 0.771 & 0.960 & 250 & 1.0 \\ \hline
  $\theta$ & 7.215 & 0.960 & 7.229 & 0.944 & 250 & 3.0 \\ \hline
  $\nu$ & 0.118 & 0.996 & 0.180 & 0.616 & 157 & 1.0 \\ \hline\hline
  $\lambda$ & 2.461 & 0.944 & 2.318 & 0.944 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.735 & 0.952 & 0.775 & 0.944 & 250 & 1.0 \\ \hline
  $\theta$ & 7.570 & 0.944 & 7.328 & 0.936 & 250 & 3.0 \\ \hline
  $\nu$ & 0.119 & 0.992 & 0.182 & 0.540 & 140 & 1.0 \\ \hline\hline
  $\lambda$ & 7.733 & 0.952 & 7.469 & 0.956 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.736 & 0.956 & 0.788 & 0.956 & 250 & 1.0 \\ \hline
  $\theta$ & 8.024 & 0.964 & 7.549 & 0.968 & 250 & 3.0 \\ \hline
  $\nu$ & 0.120 & 0.996 & 0.194 & 0.504 & 130 & 1.0 \\ \hline\hline
  $\lambda$ & 0.270 & 0.952 & 0.274 & 0.956 & 250 & 0.5 \\ \hline
  $\alpha$ & 2.242 & 0.968 & 2.331 & 0.956 & 250 & 3.0 \\ \hline
  $\theta$ & 8.277 & 0.964 & 8.599 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 0.117 & 1.000& 0.190 & 0.480 & 122 & 1.0 \\ \hline\hline
  $\lambda$ & 0.550 & 0.948 & 0.550 & 0.936 & 250 & 1.0 \\ \hline
  $\alpha$ & 2.258 & 0.964 & 2.332 & 0.936 & 250 & 3.0 \\ \hline
  $\theta$ & 8.662 & 0.964 & 8.896 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 0.127 & 0.996 & 0.213 & 0.476 & 127 & 1.0 \\ \hline\hline
  $\lambda$ & 1.661 & 0.928 & 1.614 & 0.940 & 250 & 3.0 \\ \hline
  $\alpha$ & 2.262 & 0.968 & 2.372 & 0.940 & 250 & 3.0 \\ \hline
  $\theta$ & 8.518 & 0.944 & 8.076 & 0.940 & 250 & 3.0 \\ \hline
  $\nu$ & 0.126 & 0.992 & 0.198 & 0.560 & 145 & 1.0 \\ \hline\hline
  $\lambda$ & 3.761 & 0.936 & 4.828 & 0.940 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.520 & 0.972 & 0.486 & 0.940 & 250 & 0.5 \\ \hline
  $\theta$ & 0.679 & 0.936 & 0.707 & 0.944 & 250 & 0.5 \\ \hline
  $\nu$ & 0.790 & 0.936 & 0.802 & 0.944 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 9.172 & 0.956 & 11.478 & 0.936 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.515 & 0.956 & 0.483 & 0.936 & 250 & 0.5 \\ \hline
  $\theta$ & 0.689 & 0.928 & 0.719 & 0.948 & 250 & 0.5 \\ \hline
  $\nu$ & 0.798 & 0.948 & 0.810 & 0.964 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 31.964 & 0.948 & 33.826 & 0.940 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.516 & 0.956 & 0.485 & 0.940 & 250 & 0.5 \\ \hline
  $\theta$ & 0.696 & 0.916 & 0.720 & 0.924 & 250 & 0.5 \\ \hline
  $\nu$ & 0.783 & 0.956 & 0.795 & 0.964 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.046 & 0.940 & 1.089 & 0.904 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.041 & 0.920 & 0.977 & 0.904 & 250 & 1.0 \\ \hline
  $\theta$ & 0.696 & 0.928 & 0.723 & 0.916 & 250 & 0.5 \\ \hline
  $\nu$ & 0.799 & 0.944 & 0.812 & 0.948 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.814 & 0.936 & 1.842 & 0.948 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.034 & 0.968 & 0.967 & 0.948 & 250 & 1.0 \\ \hline
  $\theta$ & 0.665 & 0.932 & 0.694 & 0.960 & 250 & 0.5 \\ \hline
  $\nu$ & 0.791 & 0.936 & 0.804 & 0.960 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 5.354 & 0.936 & 5.404 & 0.936 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.053 & 0.960 & 0.989 & 0.936 & 250 & 1.0 \\ \hline
  $\theta$ & 0.667 & 0.924 & 0.695 & 0.944 & 250 & 0.5 \\ \hline
  $\nu$ & 0.803 & 0.968 & 0.815 & 0.956 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.227 & 0.964 & 0.230 & 0.964 & 250 & 0.5 \\ \hline
  $\alpha$ & 3.120 & 0.976 & 2.904 & 0.964 & 250 & 3.0 \\ \hline
  $\theta$ & 0.682 & 0.956 & 0.715 & 0.968 & 250 & 0.5 \\ \hline
  $\nu$ & 0.791 & 0.944 & 0.804 & 0.952 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.450 & 0.952 & 0.456 & 0.968 & 250 & 1.0 \\ \hline
  $\alpha$ & 3.134 & 0.972 & 2.930 & 0.968 & 250 & 3.0 \\ \hline
  $\theta$ & 0.676 & 0.944 & 0.704 & 0.972 & 250 & 0.5 \\ \hline
  $\nu$ & 0.803 & 0.956 & 0.818 & 0.940 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.412 & 0.964 & 1.430 & 0.948 & 250 & 3.0 \\ \hline
  $\alpha$ & 3.072 & 0.944 & 2.885 & 0.948 & 250 & 3.0 \\ \hline
  $\theta$ & 0.705 & 0.944 & 0.734 & 0.940 & 250 & 0.5 \\ \hline
  $\nu$ & 0.795 & 0.928 & 0.809 & 0.936 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 4.630 & 0.964 & 6.291 & 0.968 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.447 & 0.976 & 0.426 & 0.968 & 250 & 0.5 \\ \hline
  $\theta$ & 1.654 & 0.952 & 1.693 & 0.960 & 250 & 1.0 \\ \hline
  $\nu$ & 0.802 & 0.952 & 0.816 & 0.952 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 8.157 & 0.956 & 11.014 & 0.956 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.458 & 0.976 & 0.436 & 0.956 & 250 & 0.5 \\ \hline
  $\theta$ & 1.595 & 0.948 & 1.631 & 0.972 & 250 & 1.0 \\ \hline
  $\nu$ & 0.798 & 0.916 & 0.810 & 0.936 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 25.013 & 0.932 & 34.692 & 0.948 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.451 & 0.972 & 0.430 & 0.948 & 250 & 0.5 \\ \hline
  $\theta$ & 1.675 & 0.928 & 1.712 & 0.920 & 250 & 1.0 \\ \hline
  $\nu$ & 0.804 & 0.920 & 0.818 & 0.908 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.05 & 0.960 & 1.073 & 0.956 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.879 & 0.980 & 0.840 & 0.956 & 250 & 1.0 \\ \hline
  $\theta$ & 1.730 & 0.956 & 1.753 & 0.960 & 250 & 1.0 \\ \hline
  $\nu$ & 0.791 & 0.956 & 0.804 & 0.964 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 2.183 & 0.956 & 2.245 & 0.960 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.887 & 0.968 & 0.846 & 0.960 & 250 & 1.0 \\ \hline
  $\theta$ & 1.787 & 0.944 & 1.808 & 0.936 & 250 & 1.0 \\ \hline
  $\nu$ & 0.787 & 0.944 & 0.802 & 0.956 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 5.701 & 0.940 & 5.930 & 0.936 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.922 & 0.976 & 0.878 & 0.936 & 250 & 1.0 \\ \hline
  $\theta$ & 1.633 & 0.924 & 1.679 & 0.944 & 250 & 1.0 \\ \hline
  $\nu$ & 0.795 & 0.960 & 0.811 & 0.972 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.227 & 0.944 & 0.229 & 0.956 & 250 & 0.5 \\ \hline
  $\alpha$ & 2.740 & 0.980 & 2.616 & 0.956 & 250 & 3.0 \\ \hline
  $\theta$ & 1.577 & 0.924 & 1.600 & 0.960 & 250 & 1.0 \\ \hline
  $\nu$ & 0.802 & 0.936 & 0.816 & 0.936 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.495 & 0.944 & 0.501 & 0.944 & 250 & 1.0 \\ \hline
  $\alpha$ & 2.635 & 0.964 & 2.517 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 1.783 & 0.940 & 1.806 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 0.787 & 0.944 & 0.800 & 0.940 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.453 & 0.944 & 1.471 & 0.944 & 250 & 3.0 \\ \hline
  $\alpha$ & 2.660 & 0.976 & 2.549 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 1.743 & 0.932 & 1.775 & 0.920 & 250 & 1.0 \\ \hline
  $\nu$ & 0.787 & 0.940 & 0.801 & 0.940 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 11.245 & 0.964 & 16.381 & 0.948 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.417 & 0.940 & 0.408 & 0.948 & 250 & 0.5 \\ \hline
  $\theta$ & 9.680 & 0.968 & 10.510 & 0.932 & 250 & 3.0 \\ \hline
  $\nu$ & 0.797 & 0.960 & 0.813 & 0.968 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 14.024 & 0.940 & 20.867 & 0.932 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.442 & 0.948 & 0.434 & 0.932 & 250 & 0.5 \\ \hline
  $\theta$ & 8.183 & 0.920 & 8.847 & 0.920 & 250 & 3.0 \\ \hline
  $\nu$ & 0.784 & 0.928 & 0.796 & 0.940 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 43.521 & 0.968 & 62.684 & 0.964 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.424 & 0.968 & 0.414 & 0.964 & 250 & 0.5 \\ \hline
  $\theta$ & 8.900 & 0.964 & 9.638 & 0.968 & 250 & 3.0 \\ \hline
  $\nu$ & 0.786 & 0.952 & 0.801 & 0.964 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.445 & 0.948 & 1.711 & 0.964 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.870 & 0.968 & 0.851 & 0.964 & 250 & 1.0 \\ \hline
  $\theta$ & 8.814 & 0.960 & 9.810 & 0.960 & 250 & 3.0 \\ \hline
  $\nu$ & 0.791 & 0.952 & 0.805 & 0.956 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 3.100 & 0.956 & 3.544 & 0.944 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.874 & 0.964 & 0.860 & 0.944 & 250 & 1.0 \\ \hline
  $\theta$ & 9.289 & 0.940 & 10.231 & 0.940 & 250 & 3.0 \\ \hline
  $\nu$ & 0.787 & 0.964 & 0.802 & 0.968 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 9.048 & 0.960 & 10.132 & 0.956 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.856 & 0.968 & 0.839 & 0.956 & 250 & 1.0 \\ \hline
  $\theta$ & 9.163 & 0.968 & 10.139 & 0.972 & 250 & 3.0 \\ \hline
  $\nu$ & 0.795 & 0.924 & 0.808 & 0.948 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.318 & 0.944 & 0.321 & 0.948 & 250 & 0.5 \\ \hline
  $\alpha$ & 2.614 & 0.956 & 2.565 & 0.948 & 250 & 3.0 \\ \hline
  $\theta$ & 10.068 & 0.956 & 11.265 & 0.948 & 250 & 3.0 \\ \hline
  $\nu$ & 0.801 & 0.952 & 0.817 & 0.952 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.685 & 0.984 & 0.689 & 0.968 & 250 & 1.0 \\ \hline
  $\alpha$ & 2.534 & 0.972 & 2.494 & 0.968 & 250 & 3.0 \\ \hline
  $\theta$ & 11.350 & 0.968 & 12.878 & 0.948 & 250 & 3.0 \\ \hline
  $\nu$ & 0.788 & 0.960 & 0.804 & 0.968 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 2.135 & 0.968 & 2.141 & 0.944 & 250 & 3.0 \\ \hline
  $\alpha$ & 2.547 & 0.964 & 2.501 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 12.469 & 0.960 & 14.199 & 0.932 & 250 & 3.0 \\ \hline
  $\nu$ & 0.792 & 0.936 & 0.807 & 0.944 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 4.938 & 0.960 & 6.561 & 0.948 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.510 & 0.964 & 0.478 & 0.948 & 250 & 0.5 \\ \hline
  $\theta$ & 0.716 & 0.964 & 0.740 & 0.952 & 250 & 0.5 \\ \hline
  $\nu$ & 1.288 & 0.960 & 1.308 & 0.968 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 7.576 & 0.956 & 9.141 & 0.960 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.514 & 0.972 & 0.481 & 0.960 & 250 & 0.5 \\ \hline
  $\theta$ & 0.712 & 0.948 & 0.733 & 0.960 & 250 & 0.5 \\ \hline
  $\nu$ & 1.286 & 0.940 & 1.306 & 0.948 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 22.928 & 0.928 & 27.830 & 0.924 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.527 & 0.944 & 0.495 & 0.924 & 250 & 0.5 \\ \hline
  $\theta$ & 0.693 & 0.916 & 0.711 & 0.940 & 250 & 0.5 \\ \hline
  $\nu$ & 1.317 & 0.952 & 1.340 & 0.952 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.997 & 0.956 & 0.994 & 0.952 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.035 & 0.960 & 0.975 & 0.952 & 250 & 1.0 \\ \hline
  $\theta$ & 0.723 & 0.952 & 0.745 & 0.944 & 250 & 0.5 \\ \hline
  $\nu$ & 1.273 & 0.928 & 1.295 & 0.944 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 1.913 & 0.940 & 1.896 & 0.932 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.035 & 0.940 & 0.976 & 0.932 & 250 & 1.0 \\ \hline
  $\theta$ & 0.698 & 0.940 & 0.718 & 0.952 & 250 & 0.5 \\ \hline
  $\nu$ & 1.297 & 0.956 & 1.317 & 0.948 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 6.244 & 0.960 & 6.227 & 0.948 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.022 & 0.952 & 0.965 & 0.948 & 250 & 1.0 \\ \hline
  $\theta$ & 0.741 & 0.948 & 0.758 & 0.924 & 250 & 0.5 \\ \hline
  $\nu$ & 1.283 & 0.932 & 1.302 & 0.952 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.242 & 0.952 & 0.243 & 0.944 & 250 & 0.5 \\ \hline
  $\alpha$ & 3.093 & 0.956 & 2.909 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 0.717 & 0.936 & 0.737 & 0.944 & 250 & 0.5 \\ \hline
  $\nu$ & 1.276 & 0.928 & 1.295 & 0.944 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.479 & 0.968 & 0.482 & 0.952 & 250 & 1.0 \\ \hline
  $\alpha$ & 3.118 & 0.964 & 2.929 & 0.952 & 250 & 3.0 \\ \hline
  $\theta$ & 0.721 & 0.952 & 0.742 & 0.956 & 250 & 0.5 \\ \hline
  $\nu$ & 1.273 & 0.960 & 1.293 & 0.964 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 1.404 & 0.964 & 1.408 & 0.952 & 250 & 3.0 \\ \hline
  $\alpha$ & 3.152 & 0.960 & 2.967 & 0.952 & 250 & 3.0 \\ \hline
  $\theta$ & 0.685 & 0.928 & 0.703 & 0.956 & 250 & 0.5 \\ \hline
  $\nu$ & 1.282 & 0.956 & 1.302 & 0.972 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 8.336 & 0.956 & 11.860 & 0.924 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.439 & 0.956 & 0.422 & 0.924 & 250 & 0.5 \\ \hline
  $\theta$ & 1.874 & 0.952 & 1.893 & 0.928 & 250 & 1.0 \\ \hline
  $\nu$ & 1.289 & 0.956 & 1.310 & 0.960 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 11.302 & 0.944 & 16.065 & 0.928 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.441 & 0.964 & 0.424 & 0.928 & 250 & 0.5 \\ \hline
  $\theta$ & 1.768 & 0.932 & 1.786 & 0.948 & 250 & 1.0 \\ \hline
  $\nu$ & 1.286 & 0.932 & 1.309 & 0.940 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 29.973 & 0.952 & 39.850 & 0.932 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.446 & 0.956 & 0.429 & 0.932 & 250 & 0.5 \\ \hline
  $\theta$ & 1.797 & 0.940 & 1.819 & 0.940 & 250 & 1.0 \\ \hline
  $\nu$ & 1.297 & 0.932 & 1.317 & 0.948 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 1.033 & 0.956 & 1.065 & 0.940 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.910 & 0.968 & 0.872 & 0.940 & 250 & 1.0 \\ \hline
  $\theta$ & 1.739 & 0.952 & 1.759 & 0.932 & 250 & 1.0 \\ \hline
  $\nu$ & 1.287 & 0.936 & 1.309 & 0.968 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 2.233 & 0.968 & 2.303 & 0.948 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.893 & 0.964 & 0.858 & 0.948 & 250 & 1.0 \\ \hline
  $\theta$ & 1.794 & 0.960 & 1.804 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 1.265 & 0.928 & 1.286 & 0.936 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 6.195 & 0.960 & 6.489 & 0.948 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.900 & 0.964 & 0.864 & 0.948 & 250 & 1.0 \\ \hline
  $\theta$ & 1.726 & 0.952 & 1.753 & 0.940 & 250 & 1.0 \\ \hline
  $\nu$ & 1.279 & 0.932 & 1.299 & 0.936 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.240 & 0.956 & 0.240 & 0.940 & 250 & 0.5 \\ \hline
  $\alpha$ & 2.738 & 0.968 & 2.620 & 0.940 & 250 & 3.0 \\ \hline
  $\theta$ & 1.738 & 0.936 & 1.739 & 0.960 & 250 & 1.0 \\ \hline
  $\nu$ & 1.289 & 0.968 & 1.310 & 0.956 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.531 & 0.968 & 0.532 & 0.940 & 250 & 1.0 \\ \hline
  $\alpha$ & 2.554 & 0.952 & 2.449 & 0.940 & 250 & 3.0 \\ \hline
  $\theta$ & 2.017 & 0.964 & 2.033 & 0.900 & 250 & 1.0 \\ \hline
  $\nu$ & 1.285 & 0.940 & 1.304 & 0.944 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 1.471 & 0.948 & 1.478 & 0.936 & 250 & 3.0 \\ \hline
  $\alpha$ & 2.659 & 0.972 & 2.546 & 0.936 & 250 & 3.0 \\ \hline
  $\theta$ & 1.778 & 0.944 & 1.792 & 0.956 & 250 & 1.0 \\ \hline
  $\nu$ & 1.281 & 0.932 & 1.300 & 0.932 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 9.409 & 0.988 & 14.212 & 0.956 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.416 & 0.972 & 0.408 & 0.956 & 250 & 0.5 \\ \hline
  $\theta$ & 9.235 & 0.968 & 9.969 & 0.940 & 250 & 3.0 \\ \hline
  $\nu$ & 1.276 & 0.940 & 1.297 & 0.944 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 16.392 & 0.980 & 23.439 & 0.976 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.417 & 0.980 & 0.410 & 0.976 & 250 & 0.5 \\ \hline
  $\theta$ & 9.395 & 0.972 & 10.156 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 1.267 & 0.944 & 1.291 & 0.940 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 38.434 & 0.948 & 55.307 & 0.940 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.434 & 0.956 & 0.427 & 0.940 & 250 & 0.5 \\ \hline
  $\theta$ & 8.540 & 0.948 & 9.173 & 0.936 & 250 & 3.0 \\ \hline
  $\nu$ & 1.269 & 0.944 & 1.291 & 0.924 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 1.572 & 0.952 & 1.794 & 0.956 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.850 & 0.972 & 0.840 & 0.956 & 250 & 1.0 \\ \hline
  $\theta$ & 9.362 & 0.964 & 10.102 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 1.291 & 0.936 & 1.313 & 0.940 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 3.404 & 0.952 & 3.957 & 0.964 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.834 & 0.968 & 0.824 & 0.964 & 250 & 1.0 \\ \hline
  $\theta$ & 10.295 & 0.956 & 11.369 & 0.952 & 250 & 3.0 \\ \hline
  $\nu$ & 1.280 & 0.956 & 1.300 & 0.948 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 10.584 & 0.960 & 12.570 & 0.960 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.843 & 0.964 & 0.827 & 0.960 & 250 & 1.0 \\ \hline
  $\theta$ & 10.328 & 0.968 & 11.516 & 0.952 & 250 & 3.0 \\ \hline
  $\nu$ & 1.279 & 0.936 & 1.304 & 0.932 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.309 & 0.964 & 0.310 & 0.964 & 250 & 0.5 \\ \hline
  $\alpha$ & 2.581 & 0.976 & 2.538 & 0.964 & 250 & 3.0 \\ \hline
  $\theta$ & 9.760 & 0.968 & 10.604 & 0.956 & 250 & 3.0 \\ \hline
  $\nu$ & 1.297 & 0.948 & 1.320 & 0.948 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.634 & 0.964 & 0.636 & 0.972 & 250 & 1.0 \\ \hline
  $\alpha$ & 2.587 & 0.964 & 2.543 & 0.972 & 250 & 3.0 \\ \hline
  $\theta$ & 10.651 & 0.968 & 11.822 & 0.952 & 250 & 3.0 \\ \hline
  $\nu$ & 1.289 & 0.936 & 1.317 & 0.944 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 1.886 & 0.960 & 1.871 & 0.948 & 250 & 3.0 \\ \hline
  $\alpha$ & 2.582 & 0.956 & 2.541 & 0.948 & 250 & 3.0 \\ \hline
  $\theta$ & 10.084 & 0.960 & 10.794 & 0.960 & 250 & 3.0 \\ \hline
  $\nu$ & 1.276 & 0.956 & 1.299 & 0.960 & 250 & 3.0 \\ \hline
    \caption{Bootstrap $z$ vs. The Bootstrap $\text{BC}_a$ Table for $n=100$}
    \label{bstab100}
\end{longtable}
\clearpage
% latex table generated in R 3.1.1 by xtable 1.7-4 package
% Tue Oct 07 13:40:39 2014
\begin{longtable}{|c|r|r|r|r|c|c|}
  \hline
      \multicolumn{7}{|c|}{$n=250$}\\ \hline
Par & Z-ci len. & Z-CI cov. & BCA-CI Len. & BCA-CI Cov. & BCA-CI Use & True \\
  \hline
$\lambda$ & 0.910 & 0.952 & 0.888 & 0.952 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.265 & 0.968 & 0.271 & 0.952 & 250 & 0.5 \\ \hline
  $\theta$ & 0.355 & 0.944 & 0.364 & 0.964 & 250 & 0.5 \\ \hline
  $\nu$ & 0.068 & 0.992 & 0.106 & 0.512 & 132 & 1.0 \\ \hline\hline
  $\lambda$ & 1.839 & 0.972 & 1.802 & 0.968 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.257 & 0.964 & 0.263 & 0.968 & 250 & 0.5 \\ \hline
  $\theta$ & 0.356 & 0.968 & 0.368 & 0.968 & 250 & 0.5 \\ \hline
  $\nu$ & 0.068 & 0.996 & 0.102 & 0.572 & 146 & 1.0 \\ \hline\hline
  $\lambda$ & 5.108 & 0.960 & 5.134 & 0.948 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.238 & 0.952 & 0.250 & 0.948 & 250 & 0.5 \\ \hline
  $\theta$ & 0.330 & 0.936 & 0.354 & 0.956 & 250 & 0.5 \\ \hline
  $\nu$ & 0.071 & 0.988 & 0.101 & 0.536 & 143 & 1.0 \\ \hline\hline
  $\lambda$ & 0.365 & 0.920 & 0.374 & 0.948 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.542 & 0.960 & 0.555 & 0.948 & 250 & 1.0 \\ \hline
  $\theta$ & 0.353 & 0.928 & 0.366 & 0.948 & 250 & 0.5 \\ \hline
  $\nu$ & 0.071 & 0.984 & 0.109 & 0.500 & 133 & 1.0 \\ \hline\hline
  $\lambda$ & 0.747 & 0.916 & 0.766 & 0.932 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.539 & 0.948 & 0.548 & 0.932 & 250 & 1.0 \\ \hline
  $\theta$ & 0.359 & 0.904 & 0.373 & 0.944 & 250 & 0.5 \\ \hline
  $\nu$ & 0.07 & 0.988 & 0.109 & 0.532 & 138 & 1.0 \\ \hline\hline
  $\lambda$ & 2.077 & 0.976 & 2.208 & 0.944 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.495 & 0.972 & 0.517 & 0.944 & 250 & 1.0 \\ \hline
  $\theta$ & 0.331 & 0.952 & 0.357 & 0.976 & 250 & 0.5 \\ \hline
  $\nu$ & 0.067 & 0.996 & 0.101 & 0.532 & 136 & 1.0 \\ \hline\hline
  $\lambda$ & 0.116 & 0.948 & 0.122 & 0.960 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.533 & 0.968 & 1.589 & 0.960 & 250 & 3.0 \\ \hline
  $\theta$ & 0.356 & 0.956 & 0.376 & 0.976 & 250 & 0.5 \\ \hline
  $\nu$ & 0.072 & 0.988 & 0.109 & 0.560 & 150 & 1.0 \\ \hline\hline
  $\lambda$ & 0.234 & 0.920 & 0.245 & 0.924 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.546 & 0.944 & 1.579 & 0.924 & 250 & 3.0 \\ \hline
  $\theta$ & 0.361 & 0.912 & 0.383 & 0.956 & 250 & 0.5 \\ \hline
  $\nu$ & 0.069 & 0.996 & 0.110& 0.504 & 130 & 1.0 \\ \hline\hline
  $\lambda$ & 0.694 & 0.916 & 0.726 & 0.956 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.585 & 0.948 & 1.605 & 0.956 & 250 & 3.0 \\ \hline
  $\theta$ & 0.357 & 0.904 & 0.382 & 0.948 & 250 & 0.5 \\ \hline
  $\nu$ & 0.069 & 0.996 & 0.107 & 0.540 & 137 & 1.0 \\ \hline\hline
  $\lambda$ & 0.948 & 0.948 & 0.884 & 0.952 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.235 & 0.960 & 0.247 & 0.952 & 250 & 0.5 \\ \hline
  $\theta$ & 0.785 & 0.940 & 0.796 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 0.068 & 0.992 & 0.099 & 0.568 & 148 & 1.0 \\ \hline\hline
  $\lambda$ & 2.043 & 0.960 & 1.938 & 0.964 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.226 & 0.960 & 0.239 & 0.964 & 250 & 0.5 \\ \hline
  $\theta$ & 0.807 & 0.976 & 0.824 & 0.964 & 250 & 1.0 \\ \hline
  $\nu$ & 0.068 & 0.988 & 0.107 & 0.464 & 125 & 1.0 \\ \hline\hline
  $\lambda$ & 5.897 & 0.948 & 5.898 & 0.932 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.223 & 0.936 & 0.233 & 0.932 & 250 & 0.5 \\ \hline
  $\theta$ & 0.775 & 0.940 & 0.809 & 0.932 & 250 & 1.0 \\ \hline
  $\nu$ & 0.069 & 0.992 & 0.104 & 0.572 & 146 & 1.0 \\ \hline\hline
  $\lambda$ & 0.403 & 0.948 & 0.396 & 0.928 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.468 & 0.952 & 0.489 & 0.928 & 250 & 1.0 \\ \hline
  $\theta$ & 0.812 & 0.936 & 0.826 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 0.069 & 0.992 & 0.103 & 0.608 & 155 & 1.0 \\ \hline\hline
  $\lambda$ & 0.777 & 0.960 & 0.767 & 0.932 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.474 & 0.960 & 0.494 & 0.932 & 250 & 1.0 \\ \hline
  $\theta$ & 0.798 & 0.948 & 0.807 & 0.956 & 250 & 1.0 \\ \hline
  $\nu$ & 0.068 & 0.992 & 0.103 & 0.552 & 143 & 1.0 \\ \hline\hline
  $\lambda$ & 2.396 & 0.952 & 2.461 & 0.944 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.449 & 0.948 & 0.470 & 0.944 & 250 & 1.0 \\ \hline
  $\theta$ & 0.802 & 0.940 & 0.830 & 0.948 & 250 & 1.0 \\ \hline
  $\nu$ & 0.071 & 0.988 & 0.112 & 0.508 & 130 & 1.0 \\ \hline\hline
  $\lambda$ & 0.119 & 0.956 & 0.127 & 0.944 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.392 & 0.936 & 1.436 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 0.812 & 0.948 & 0.863 & 0.968 & 250 & 1.0 \\ \hline
  $\nu$ & 0.067 & 1.000& 0.104 & 0.532 & 136 & 1.0 \\ \hline\hline
  $\lambda$ & 0.243 & 0.936 & 0.256 & 0.956 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.401 & 0.956 & 1.448 & 0.956 & 250 & 3.0 \\ \hline
  $\theta$ & 0.829 & 0.936 & 0.875 & 0.948 & 250 & 1.0 \\ \hline
  $\nu$ & 0.071 & 0.992 & 0.111 & 0.536 & 138 & 1.0 \\ \hline\hline
  $\lambda$ & 0.718 & 0.964 & 0.748 & 0.964 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.413 & 0.952 & 1.467 & 0.964 & 250 & 3.0 \\ \hline
  $\theta$ & 0.815 & 0.960 & 0.848 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 0.069 & 0.996 & 0.105 & 0.520 & 133 & 1.0 \\ \hline\hline
  $\lambda$ & 1.241 & 0.956 & 1.152 & 0.940 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.216 & 0.944 & 0.233 & 0.940 & 250 & 0.5 \\ \hline
  $\theta$ & 3.262 & 0.956 & 3.345 & 0.952 & 250 & 3.0 \\ \hline
  $\nu$ & 0.068 & 0.996 & 0.096 & 0.636 & 162 & 1.0 \\ \hline\hline
  $\lambda$ & 2.908 & 0.968 & 2.685 & 0.964 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.211 & 0.940 & 0.229 & 0.964 & 250 & 0.5 \\ \hline
  $\theta$ & 3.551 & 0.964 & 3.553 & 0.952 & 250 & 3.0 \\ \hline
  $\nu$ & 0.068 & 0.996 & 0.095 & 0.624 & 161 & 1.0 \\ \hline\hline
  $\lambda$ & 7.478 & 0.940 & 7.210 & 0.960 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.217 & 0.968 & 0.235 & 0.960 & 250 & 0.5 \\ \hline
  $\theta$ & 3.269 & 0.956 & 3.252 & 0.956 & 250 & 3.0 \\ \hline
  $\nu$ & 0.069 & 0.996 & 0.104 & 0.548 & 141 & 1.0 \\ \hline\hline
  $\lambda$ & 0.477 & 0.972 & 0.476 & 0.976 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.429 & 0.968 & 0.455 & 0.976 & 250 & 1.0 \\ \hline
  $\theta$ & 3.364 & 0.968 & 3.505 & 0.968 & 250 & 3.0 \\ \hline
  $\nu$ & 0.072 & 0.996 & 0.1 & 0.608 & 158 & 1.0 \\ \hline\hline
  $\lambda$ & 0.917 & 0.944 & 0.878 & 0.964 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.437 & 0.964 & 0.467 & 0.964 & 250 & 1.0 \\ \hline
  $\theta$ & 3.254 & 0.944 & 3.304 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 0.072 & 0.988 & 0.108 & 0.532 & 140 & 1.0 \\ \hline\hline
  $\lambda$ & 2.987 & 0.972 & 2.940 & 0.972 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.434 & 0.976 & 0.468 & 0.972 & 250 & 1.0 \\ \hline
  $\theta$ & 3.550 & 0.968 & 3.569 & 0.960 & 250 & 3.0 \\ \hline
  $\nu$ & 0.071 & 1.000& 0.112 & 0.500 & 129 & 1.0 \\ \hline\hline
  $\lambda$ & 0.137 & 0.948 & 0.141 & 0.944 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.331 & 0.948 & 1.416 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 3.370 & 0.952 & 3.463 & 0.960 & 250 & 3.0 \\ \hline
  $\nu$ & 0.068 & 0.992 & 0.105 & 0.532 & 136 & 1.0 \\ \hline\hline
  $\lambda$ & 0.280 & 0.964 & 0.287 & 0.960 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.348 & 0.952 & 1.424 & 0.960 & 250 & 3.0 \\ \hline
  $\theta$ & 3.432 & 0.980 & 3.454 & 0.972 & 250 & 3.0 \\ \hline
  $\nu$ & 0.073 & 0.980 & 0.113 & 0.528 & 140 & 1.0 \\ \hline\hline
  $\lambda$ & 0.871 & 0.952 & 0.884 & 0.936 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.328 & 0.940 & 1.421 & 0.936 & 250 & 3.0 \\ \hline
  $\theta$ & 3.680 & 0.956 & 3.638 & 0.944 & 250 & 3.0 \\ \hline
  $\nu$ & 0.073 & 0.992 & 0.106 & 0.624 & 163 & 1.0 \\ \hline\hline
  $\lambda$ & 1.121 & 0.940 & 1.119 & 0.972 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.278 & 0.968 & 0.272 & 0.972 & 250 & 0.5 \\ \hline
  $\theta$ & 0.384 & 0.964 & 0.391 & 0.960 & 250 & 0.5 \\ \hline
  $\nu$ & 0.499 & 0.940 & 0.503 & 0.960 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.995 & 0.948 & 2.005 & 0.948 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.283 & 0.964 & 0.277 & 0.948 & 250 & 0.5 \\ \hline
  $\theta$ & 0.374 & 0.940 & 0.383 & 0.952 & 250 & 0.5 \\ \hline
  $\nu$ & 0.498 & 0.948 & 0.503 & 0.960 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 7.207 & 0.968 & 7.205 & 0.936 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.272 & 0.944 & 0.266 & 0.936 & 250 & 0.5 \\ \hline
  $\theta$ & 0.396 & 0.944 & 0.403 & 0.936 & 250 & 0.5 \\ \hline
  $\nu$ & 0.499 & 0.940 & 0.504 & 0.952 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.446 & 0.956 & 0.449 & 0.948 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.549 & 0.960 & 0.537 & 0.948 & 250 & 1.0 \\ \hline
  $\theta$ & 0.387 & 0.952 & 0.395 & 0.952 & 250 & 0.5 \\ \hline
  $\nu$ & 0.493 & 0.952 & 0.498 & 0.952 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.856 & 0.980 & 0.864 & 0.960 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.563 & 0.964 & 0.550 & 0.960 & 250 & 1.0 \\ \hline
  $\theta$ & 0.381 & 0.956 & 0.389 & 0.964 & 250 & 0.5 \\ \hline
  $\nu$ & 0.493 & 0.920 & 0.498 & 0.928 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 2.718 & 0.964 & 2.737 & 0.936 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.544 & 0.936 & 0.531 & 0.936 & 250 & 1.0 \\ \hline
  $\theta$ & 0.390 & 0.964 & 0.398 & 0.956 & 250 & 0.5 \\ \hline
  $\nu$ & 0.497 & 0.936 & 0.502 & 0.960 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.135 & 0.968 & 0.137 & 0.936 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.627 & 0.948 & 1.592 & 0.936 & 250 & 3.0 \\ \hline
  $\theta$ & 0.396 & 0.972 & 0.404 & 0.948 & 250 & 0.5 \\ \hline
  $\nu$ & 0.496 & 0.928 & 0.500 & 0.940 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.260 & 0.940 & 0.263 & 0.940 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.689 & 0.952 & 1.647 & 0.940 & 250 & 3.0 \\ \hline
  $\theta$ & 0.378 & 0.944 & 0.385 & 0.944 & 250 & 0.5 \\ \hline
  $\nu$ & 0.500 & 0.956 & 0.503 & 0.948 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.796 & 0.948 & 0.803 & 0.944 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.673 & 0.956 & 1.632 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 0.383 & 0.948 & 0.390 & 0.960 & 250 & 0.5 \\ \hline
  $\nu$ & 0.499 & 0.940 & 0.503 & 0.932 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.218 & 0.988 & 1.256 & 0.972 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.245 & 0.960 & 0.242 & 0.972 & 250 & 0.5 \\ \hline
  $\theta$ & 0.897 & 0.980 & 0.913 & 0.964 & 250 & 1.0 \\ \hline
  $\nu$ & 0.497 & 0.956 & 0.503 & 0.960 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 2.247 & 0.952 & 2.308 & 0.952 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.249 & 0.956 & 0.246 & 0.952 & 250 & 0.5 \\ \hline
  $\theta$ & 0.869 & 0.964 & 0.882 & 0.956 & 250 & 1.0 \\ \hline
  $\nu$ & 0.498 & 0.956 & 0.503 & 0.972 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 6.655 & 0.940 & 6.716 & 0.968 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.247 & 0.988 & 0.244 & 0.968 & 250 & 0.5 \\ \hline
  $\theta$ & 0.869 & 0.964 & 0.881 & 0.976 & 250 & 1.0 \\ \hline
  $\nu$ & 0.501 & 0.932 & 0.506 & 0.932 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.427 & 0.964 & 0.428 & 0.968 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.501 & 0.972 & 0.496 & 0.968 & 250 & 1.0 \\ \hline
  $\theta$ & 0.853 & 0.964 & 0.862 & 0.964 & 250 & 1.0 \\ \hline
  $\nu$ & 0.498 & 0.964 & 0.503 & 0.976 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.871 & 0.948 & 0.874 & 0.932 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.501 & 0.948 & 0.495 & 0.932 & 250 & 1.0 \\ \hline
  $\theta$ & 0.862 & 0.940 & 0.873 & 0.936 & 250 & 1.0 \\ \hline
  $\nu$ & 0.501 & 0.960 & 0.506 & 0.960 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 2.651 & 0.944 & 2.664 & 0.928 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.500 & 0.948 & 0.494 & 0.928 & 250 & 1.0 \\ \hline
  $\theta$ & 0.867 & 0.936 & 0.878 & 0.936 & 250 & 1.0 \\ \hline
  $\nu$ & 0.494 & 0.940 & 0.499 & 0.952 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.131 & 0.956 & 0.132 & 0.940 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.504 & 0.952 & 1.484 & 0.940 & 250 & 3.0 \\ \hline
  $\theta$ & 0.865 & 0.948 & 0.878 & 0.956 & 250 & 1.0 \\ \hline
  $\nu$ & 0.493 & 0.956 & 0.500 & 0.956 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.268 & 0.960 & 0.271 & 0.952 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.476 & 0.968 & 1.461 & 0.952 & 250 & 3.0 \\ \hline
  $\theta$ & 0.886 & 0.956 & 0.898 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 0.494 & 0.948 & 0.499 & 0.960 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.794 & 0.968 & 0.802 & 0.964 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.482 & 0.976 & 1.464 & 0.964 & 250 & 3.0 \\ \hline
  $\theta$ & 0.868 & 0.948 & 0.880 & 0.976 & 250 & 1.0 \\ \hline
  $\nu$ & 0.498 & 0.964 & 0.502 & 0.956 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.728 & 0.956 & 1.956 & 0.940 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.250 & 0.936 & 0.249 & 0.940 & 250 & 0.5 \\ \hline
  $\theta$ & 4.064 & 0.964 & 4.058 & 0.940 & 250 & 3.0 \\ \hline
  $\nu$ & 0.494 & 0.924 & 0.499 & 0.940 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 3.395 & 0.948 & 3.774 & 0.940 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.252 & 0.948 & 0.251 & 0.940 & 250 & 0.5 \\ \hline
  $\theta$ & 4.022 & 0.952 & 4.032 & 0.944 & 250 & 3.0 \\ \hline
  $\nu$ & 0.503 & 0.960 & 0.508 & 0.952 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 10.297 & 0.956 & 11.452 & 0.916 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.251 & 0.916 & 0.250 & 0.916 & 250 & 0.5 \\ \hline
  $\theta$ & 4.120 & 0.948 & 4.132 & 0.940 & 250 & 3.0 \\ \hline
  $\nu$ & 0.492 & 0.940 & 0.497 & 0.952 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.560 & 0.964 & 0.559 & 0.956 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.505 & 0.968 & 0.502 & 0.956 & 250 & 1.0 \\ \hline
  $\theta$ & 3.914 & 0.948 & 3.905 & 0.952 & 250 & 3.0 \\ \hline
  $\nu$ & 0.496 & 0.940 & 0.501 & 0.940 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.169 & 0.956 & 1.180 & 0.960 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.503 & 0.960 & 0.500 & 0.960 & 250 & 1.0 \\ \hline
  $\theta$ & 4.093 & 0.968 & 4.097 & 0.944 & 250 & 3.0 \\ \hline
  $\nu$ & 0.500 & 0.960 & 0.505 & 0.968 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 3.537 & 0.968 & 3.564 & 0.952 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.501 & 0.960 & 0.499 & 0.952 & 250 & 1.0 \\ \hline
  $\theta$ & 4.057 & 0.956 & 4.069 & 0.932 & 250 & 3.0 \\ \hline
  $\nu$ & 0.500 & 0.964 & 0.505 & 0.960 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.159 & 0.968 & 0.161 & 0.980 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.507 & 0.984 & 1.499 & 0.980 & 250 & 3.0 \\ \hline
  $\theta$ & 3.957 & 0.968 & 3.963 & 0.976 & 250 & 3.0 \\ \hline
  $\nu$ & 0.496 & 0.944 & 0.500 & 0.952 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.329 & 0.976 & 0.332 & 0.964 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.491 & 0.976 & 1.485 & 0.964 & 250 & 3.0 \\ \hline
  $\theta$ & 4.117 & 0.972 & 4.116 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 0.501 & 0.956 & 0.506 & 0.964 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 0.960 & 0.964 & 0.968 & 0.948 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.508 & 0.956 & 1.506 & 0.948 & 250 & 3.0 \\ \hline
  $\theta$ & 3.981 & 0.956 & 3.960 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 0.493 & 0.948 & 0.499 & 0.956 & 250 & 2.0 \\ \hline\hline
  $\lambda$ & 1.126 & 0.948 & 1.118 & 0.964 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.279 & 0.972 & 0.273 & 0.964 & 250 & 0.5 \\ \hline
  $\theta$ & 0.392 & 0.944 & 0.397 & 0.968 & 250 & 0.5 \\ \hline
  $\nu$ & 0.802 & 0.956 & 0.811 & 0.964 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 2.424 & 0.944 & 2.441 & 0.944 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.278 & 0.948 & 0.273 & 0.944 & 250 & 0.5 \\ \hline
  $\theta$ & 0.402 & 0.944 & 0.409 & 0.940 & 250 & 0.5 \\ \hline
  $\nu$ & 0.807 & 0.972 & 0.814 & 0.968 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 6.902 & 0.972 & 6.860 & 0.964 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.278 & 0.976 & 0.273 & 0.964 & 250 & 0.5 \\ \hline
  $\theta$ & 0.394 & 0.956 & 0.400 & 0.976 & 250 & 0.5 \\ \hline
  $\nu$ & 0.813 & 0.956 & 0.820 & 0.964 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.453 & 0.968 & 0.455 & 0.968 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.552 & 0.980 & 0.542 & 0.968 & 250 & 1.0 \\ \hline
  $\theta$ & 0.399 & 0.952 & 0.405 & 0.952 & 250 & 0.5 \\ \hline
  $\nu$ & 0.811 & 0.944 & 0.816 & 0.936 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.929 & 0.972 & 0.935 & 0.952 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.554 & 0.968 & 0.545 & 0.952 & 250 & 1.0 \\ \hline
  $\theta$ & 0.397 & 0.960 & 0.405 & 0.968 & 250 & 0.5 \\ \hline
  $\nu$ & 0.799 & 0.928 & 0.805 & 0.932 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 2.721 & 0.952 & 2.729 & 0.928 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.557 & 0.952 & 0.545 & 0.928 & 250 & 1.0 \\ \hline
  $\theta$ & 0.396 & 0.928 & 0.402 & 0.932 & 250 & 0.5 \\ \hline
  $\nu$ & 0.805 & 0.956 & 0.814 & 0.960 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.139 & 0.960 & 0.140 & 0.956 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.654 & 0.972 & 1.622 & 0.956 & 250 & 3.0 \\ \hline
  $\theta$ & 0.396 & 0.952 & 0.403 & 0.972 & 250 & 0.5 \\ \hline
  $\nu$ & 0.808 & 0.956 & 0.815 & 0.964 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.278 & 0.960 & 0.280 & 0.932 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.655 & 0.948 & 1.623 & 0.932 & 250 & 3.0 \\ \hline
  $\theta$ & 0.402 & 0.932 & 0.409 & 0.936 & 250 & 0.5 \\ \hline
  $\nu$ & 0.808 & 0.944 & 0.814 & 0.948 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.832 & 0.964 & 0.836 & 0.948 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.665 & 0.968 & 1.632 & 0.948 & 250 & 3.0 \\ \hline
  $\theta$ & 0.398 & 0.944 & 0.403 & 0.948 & 250 & 0.5 \\ \hline
  $\nu$ & 0.806 & 0.940 & 0.814 & 0.948 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 1.196 & 0.948 & 1.223 & 0.940 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.248 & 0.940 & 0.246 & 0.940 & 250 & 0.5 \\ \hline
  $\theta$ & 0.886 & 0.940 & 0.892 & 0.920 & 250 & 1.0 \\ \hline
  $\nu$ & 0.811 & 0.956 & 0.820 & 0.956 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 2.398 & 0.964 & 2.421 & 0.952 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.245 & 0.960 & 0.243 & 0.952 & 250 & 0.5 \\ \hline
  $\theta$ & 0.894 & 0.964 & 0.903 & 0.948 & 250 & 1.0 \\ \hline
  $\nu$ & 0.799 & 0.940 & 0.805 & 0.940 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 6.851 & 0.964 & 6.878 & 0.980 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.249 & 0.976 & 0.247 & 0.980 & 250 & 0.5 \\ \hline
  $\theta$ & 0.876 & 0.968 & 0.881 & 0.964 & 250 & 1.0 \\ \hline
  $\nu$ & 0.803 & 0.940 & 0.812 & 0.932 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.453 & 0.972 & 0.452 & 0.968 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.492 & 0.956 & 0.487 & 0.968 & 250 & 1.0 \\ \hline
  $\theta$ & 0.895 & 0.960 & 0.901 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 0.805 & 0.964 & 0.813 & 0.972 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.917 & 0.972 & 0.916 & 0.968 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.491 & 0.964 & 0.487 & 0.968 & 250 & 1.0 \\ \hline
  $\theta$ & 0.905 & 0.964 & 0.914 & 0.952 & 250 & 1.0 \\ \hline
  $\nu$ & 0.810 & 0.976 & 0.818 & 0.972 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 2.808 & 0.956 & 2.807 & 0.936 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.493 & 0.944 & 0.489 & 0.936 & 250 & 1.0 \\ \hline
  $\theta$ & 0.919 & 0.952 & 0.929 & 0.928 & 250 & 1.0 \\ \hline
  $\nu$ & 0.800 & 0.948 & 0.808 & 0.972 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.136 & 0.940 & 0.136 & 0.944 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.478 & 0.944 & 1.465 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 0.892 & 0.956 & 0.898 & 0.940 & 250 & 1.0 \\ \hline
  $\nu$ & 0.810 & 0.956 & 0.819 & 0.952 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.269 & 0.940 & 0.270 & 0.952 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.484 & 0.968 & 1.469 & 0.952 & 250 & 3.0 \\ \hline
  $\theta$ & 0.882 & 0.968 & 0.891 & 0.968 & 250 & 1.0 \\ \hline
  $\nu$ & 0.798 & 0.924 & 0.807 & 0.936 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.817 & 0.964 & 0.821 & 0.944 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.481 & 0.952 & 1.465 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 0.902 & 0.960 & 0.908 & 0.948 & 250 & 1.0 \\ \hline
  $\nu$ & 0.799 & 0.956 & 0.804 & 0.960 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 1.465 & 0.952 & 1.605 & 0.960 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.250 & 0.964 & 0.250 & 0.960 & 250 & 0.5 \\ \hline
  $\theta$ & 3.881 & 0.956 & 3.853 & 0.960 & 250 & 3.0 \\ \hline
  $\nu$ & 0.801 & 0.924 & 0.809 & 0.928 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 2.949 & 0.960 & 3.174 & 0.976 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.250 & 0.972 & 0.250 & 0.976 & 250 & 0.5 \\ \hline
  $\theta$ & 3.907 & 0.972 & 3.874 & 0.976 & 250 & 3.0 \\ \hline
  $\nu$ & 0.803 & 0.944 & 0.810 & 0.940 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 10.132 & 0.968 & 11.003 & 0.968 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.247 & 0.960 & 0.246 & 0.968 & 250 & 0.5 \\ \hline
  $\theta$ & 4.123 & 0.968 & 4.103 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 0.808 & 0.948 & 0.815 & 0.964 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.599 & 0.960 & 0.597 & 0.952 & 250 & 0.5 \\ \hline
  $\alpha$ & 0.494 & 0.956 & 0.493 & 0.952 & 250 & 1.0 \\ \hline
  $\theta$ & 4.180 & 0.976 & 4.143 & 0.944 & 250 & 3.0 \\ \hline
  $\nu$ & 0.806 & 0.940 & 0.814 & 0.952 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 1.237 & 0.948 & 1.223 & 0.940 & 250 & 1.0 \\ \hline
  $\alpha$ & 0.492 & 0.944 & 0.490 & 0.940 & 250 & 1.0 \\ \hline
  $\theta$ & 4.330 & 0.956 & 4.271 & 0.948 & 250 & 3.0 \\ \hline
  $\nu$ & 0.804 & 0.956 & 0.815 & 0.960 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 3.343 & 0.940 & 3.343 & 0.948 & 250 & 3.0 \\ \hline
  $\alpha$ & 0.499 & 0.952 & 0.498 & 0.948 & 250 & 1.0 \\ \hline
  $\theta$ & 3.948 & 0.936 & 3.923 & 0.964 & 250 & 3.0 \\ \hline
  $\nu$ & 0.804 & 0.940 & 0.812 & 0.936 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.168 & 0.956 & 0.169 & 0.940 & 250 & 0.5 \\ \hline
  $\alpha$ & 1.474 & 0.948 & 1.473 & 0.940 & 250 & 3.0 \\ \hline
  $\theta$ & 4.389 & 0.964 & 4.369 & 0.932 & 250 & 3.0 \\ \hline
  $\nu$ & 0.799 & 0.936 & 0.807 & 0.952 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.317 & 0.936 & 0.319 & 0.940 & 250 & 1.0 \\ \hline
  $\alpha$ & 1.509 & 0.944 & 1.510 & 0.940 & 250 & 3.0 \\ \hline
  $\theta$ & 4.040& 0.932 & 4.005 & 0.928 & 250 & 3.0 \\ \hline
  $\nu$ & 0.805 & 0.964 & 0.813 & 0.964 & 250 & 3.0 \\ \hline\hline
  $\lambda$ & 0.984 & 0.956 & 0.985 & 0.944 & 250 & 3.0 \\ \hline
  $\alpha$ & 1.485 & 0.948 & 1.483 & 0.944 & 250 & 3.0 \\ \hline
  $\theta$ & 4.214 & 0.964 & 4.173 & 0.932 & 250 & 3.0 \\ \hline
  $\nu$ & 0.805 & 0.956 & 0.811 & 0.952 & 250 & 3.0 \\ \hline
  \caption{Bootstrap $z$ vs. The Bootstrap $\text{BC}_a$ Table for $n=250$}
  \label{bstab250}
\end{longtable}
\chapter{Appendix:  Computer code}
\begin{singlespacing}
\section{Sampling Algorithms}
\subsection{Monte Carlo Gibbs Sampleer}
\begin{verbatim}
ewsample.MC <- function(n, m, binsize, lambda, alpha, theta, v)
 # n = sample size
 # m = number of iterations
 # binsize = size in MC integral
 # lambda - v = values of the parameters
{
  bins<-rexp(binsize, 1)
  x1 <- matrix(0, n, 1)
  x2 <- matrix(0, n, 1)
  condit <- function(t, x2, x1, v)
  {
    ## Defining the Monte Carlo integral 
    ## for condit cdf of bivariate exp
    ## F(x1|x2)
    m<-length(x1)
    Ftildet <- ((1/m) * sum((exp(-(x1^v + x2^v)^(1/v))
               * (x1^(v - 1) * x2^(v - 1)) 
               * ((x1^v + x2^v)^((2 - 2 * v) / v) 
               + (v - 1) * (x1^v + x2^v)^((1 - 2 * v) / v))) 
               / (exp(-x1) * exp(-x2)) * (x1<=t)))
    return(Ftildet)
  }
  mcsample <- function(u, x1, x2, v)
  {
    ## Setting to be max or min
    ## if necessary
    x1 <- sort(x1)
    if (condit(min(x1),x2,x1,v)>=u)
    {
     t <- min(x1)
    }
    else if (condit(max(x1),x2,x1,v)<=u)
    {
     t <- max(x1)
    }
    else
    {
     ## Do a bisection method
     t <- x1[which(abs(x1 - log(1-u)) == min(abs(x1 - log(1-u))))]
     R <- length(x1)
     xR <- x1[R]
     L <- 1
     xL <- x1[L]
     while ((R - L) > 1)
     {
       if (condit(t, x2, x1, v) < u)
       {
        xL <- t
        L <- which(x1 == xL)
       }
       else
       {
        xR <- t
        R <- which(x1 == xR)
       }
       t <- (xR + xL) / 2
       t <- x1[which(abs(x1 - t) == min(abs(x1 - t)))]
      }
      ## Linear Interpolation 
      t <- (xL + (xR - xL) 
           / (condit(xR, x2, x1, v)
           - condit(xL, x2, x1, v))
            * (u-condit(xL,x2,x1,v)))
     }
   return(t)
  }
 ## Sampling from bivariate exp
 ## Using MC Gibbs
 for(i in 1:n)
  {
   x1m<-vector(length=m)
   x1m[1] <- rexp(1)
   x2m<-vector(length=m)
   x2m[1] <- rexp(1) 
   for(j in 2:m)
   {
    u <- runif(1,0,1)
    x1m[j] <- mcsample(u, bins, x2m[j-1], v)
    u<-runif(1,0,1)
    x2m[j] <- mcsample(u, bins, x1m[j], v)
   }
   x1[i] <- x1m[m]
   x2[i] <- x2m[m]
  }
  x <- cbind(x1, x2)
  y <- matrix(0, n, 2)
  y[1:n, 1] <- (1 / lambda) 
               * (-log(1 - exp(-(x[1:n, 1]))^(1 / theta)))^(1 / alpha)
  y[1:n, 2] <- (1 / lambda)
                *(-log(1 - exp(-(x[1:n, 2]))^(1 / theta)))^(1 / alpha)
  return(y)
 }
 \end{verbatim}
 \subsection{Copula Gibbs Sampler}
 \begin{verbatim}
sample<-function(lambda, alpha, theta, v1, n, iter)
{
 ## Defining the conditional copula
 f <- function(w, x, v, t) 
    { 
      copula <- (exp(-((-log(w))^v + (-log(x))^v)^(1 / v)) 
       * ((-log(w))^v + (-log(x))^v)^((1 - v) / v)
       * (-log(w))^(v - 1) * (1 / w) - t)
    }
    ## Attempting 1st sample
    u <- matrix(0,n,2)
    for (i in 1:n)
    {
     u1 <- vector(, iter)
     u2 <- vector(, iter)
     u1[1] <- runif(1, 0, 1)
     u2[1] <- runif(1, 0, 1)
     for (j in 2:iter)
     {
      t <- runif(1, 0, 1)
      u1[j] <- NA
      try(u1[j] <- uniroot(f, c(0,1), w = u2[j - 1], 
                           v = v1, t = t)$root, silent = T)
      if (is.na(u1[j]))
      {
        u1[j] <- 0
      }
      t <- runif(1, 0, 1)
      u2[j] <- NA
      try(u2[j] <- uniroot(f, c(0, 1), w = u1[j], 
                           v = v1,t = t)$root, silent=T)
      if (is.na(u2[j]))
      {
        u2[j] <- 0
      }
     }
     u[i,1] <- u1[iter]
     u[i,2] <- u2[iter]
    }
    y <- (1 / lambda) * (-log(1 - u^(1 / theta)))^(1 / alpha)
    for (i in 1:n)
    {
     ## If components are bad, resample
     while (y[i, 1] == Inf || y[i, 2] == Inf || 
            y[i,1] == 0 || y[i,2] == 0)
     {
      u1 <- vector( ,iter)
      u2 <- vector( ,iter)
      u1[1] <- runif(1, 0, 1)
      u2[1] <- runif(1, 0, 1)
      for (j in 2:iter)
      {
       t <- runif(1, 0, 1)
       u1[j] <- NA
       try(u1[j] <- uniroot(f, c(0, 1), w = u2[j - 1], 
                            v = v1, t = t)$root, silent=T)
       if (is.na(u1[j]))
       {
        u1[j] <- 0
       }
       t <- runif(1,0,1)
       try(u2[j] <- uniroot(f, c(0, 1), w = u1[j],
                            v = v1, t = t)$root, silent=T)
       if (is.na(u2[j]))
       {
        u2[j] <- 0
       }
      }
      u[i,1] <- u1[iter]
      u[i,2] <- u2[iter]
      y[i, ] <- ((1 / lambda) 
                * (-log(1 - u[i, ]^(1 / theta)))^(1 / alpha))
      }
     }
     return(y)
    }

 \end{verbatim}
\section{Maximum Likelihood Estimation}
\begin{verbatim}
mmle<-function(x,lambda1,alpha1,theta1,v1)
{
   ## x = sample
   ## lambda1 - v1 = starting values of the parameters
   minusloglik<-function(par,x)
   {
     ## Defining the negative likelihood function
     n      <- length(x[, 1])
     lambda <- par[1]
     alpha  <- par[2]
     theta  <- par[3]
     v      <- par[4]

     if (lambda <= 0 || alpha <= 0 || theta <= 0 || v < 1)
    {
       ans <- Inf
     }
     else
     {  
       R1 <- -theta * log(1-exp(-(lambda * x[1:n, 1])^alpha))
       R2 <- -theta * log(1-exp(-(lambda * x[1:n, 2])^alpha))
       r1 <- (log(theta) + (theta - 1) 
             * log(1 - exp(-(lambda*x[1:n, 1])^alpha))
             - (lambda * x[1:n, 1])^alpha 
             + log(alpha) + alpha * log(lambda)
             + (alpha - 1) * log(x[1:n, 1]))
       r2 <- (log(theta) + (theta - 1)
             * log(1 - exp(-(lambda * x[1:n, 2])^alpha)) 
             - (lambda * x[1:n, 2])^alpha 
             + log(alpha) + alpha * log(lambda)
             + (alpha - 1) * log(x[1:n, 2])


       s11 <- (v-1)*sum(log(R1))+sum(R1)
       s12 <- sum(r1)
       s21 <- (v-1)*sum(log(R2))+sum(R2)
       s22 <- sum(r2)
       s3 <- sum(-(R1^(v)+R2^(v))^(1/v))
       s4 <- sum(log((v-1)*(R1^(v)+R2^(v))^((1-2*v)/v)
       +(R1^(v)+R2^(v))^((2-2*v)/v)))

       ans<- -1*(s11+s12+s21+s22+s3+s4)
      }

     return(ans)
   }

   out<-optim(c(lambda1,alpha1,theta1,v1),
        minusloglik,x=x,method="Nelder-Mead")
   ## Use optim to obatin MLEs

   return(out)

   }
\end{verbatim}
\section{Inferential Procedures}
\subsection{Mahalanobis Distance Based on the Information Matrix}
\begin{verbatim}
info.dist <- function(par,parnot,x)
{
  ## par = mles of the mvew model
  ## parnot = null parameters
  ## x =  data
  info <- matrix(0,4,4)
  fishalpha<- function(x, lambda1, alpha1, theta1, v1)
   {
   ## Partial Derivatives
   ## wrt alpha
     loglik <- expression(
               (v - 1) * (log((-theta 
               * log(1 - exp(-(lambda * x1)^alpha))))) 
               + ((-theta * log(1 - exp(-(lambda 
               * x1)^alpha)))) + ((log(theta)
               + (theta - 1) * log(1 
               - exp(-(lambda * x1)^alpha)) 
               - (lambda * x1)^alpha + log(alpha) 
               + alpha * log(lambda) 
               + (alpha - 1) * log(x1))) + (v-1) * (log((-theta 
               * log(1 - exp(-(lambda * x2)^alpha))))) 
               + ((-theta * log(1 - exp(-(lambda * x2)^alpha)))) 
               + ((log(theta) + (theta - 1) 
               * log(1 - exp(-(lambda * x2)^alpha)) 
               - (lambda * x2)^alpha 
               + log(alpha) + alpha 
               * log(lambda) + (alpha - 1) * log(x2))) + (-((-theta 
               * log(1 - exp(-(lambda * x1)^alpha)))^(v) 
               + (-theta * log(1 - exp(-(lambda 
               * x2)^alpha)))^(v))^(1 / v)) 
               + (log((v-1) * ((-theta 
               * log(1 - exp(-(lambda * x1)^alpha)))^(v) 
               + (-theta * log(1-exp(-(lambda 
               * x2)^alpha)))^(v))^((1 - 2 * v) / v) 
               + ((-theta * log(1 - exp(-(lambda 
               * x1)^alpha)))^(v) 
               + (-theta * log(1 - exp(-(lambda 
               * x2)^alpha)))^(v))^((2 - 2 * v) / v))))
    dloglike <- D(loglik, "alpha")
    dada     <- D(dloglike, "alpha")
    dadt     <- D(dloglike, "theta")
    dadv     <- D(dloglike, "v")
    

     lambda <- lambda1
     alpha  <- alpha1
     theta  <- theta1
     v      <- v1
     x1     <- x[, 1]
     x2     <- x[, 2]
     aa     <- eval(dada)
     at     <- eval(dadt)
     av     <- eval(dadv)
     ans    <- c(-1 * sum(aa), -1 * sum(at), -1 * sum(av))
    return(ans)
  }
  fishlambda<- function(x, lambda1, alpha1, theta1, v1)
   {
     ## Partial Derivatives
     ## wrt lambda
     loglik <- expression(
               (v - 1) * (log((-theta 
               * log(1 - exp(-(lambda * x1)^alpha))))) 
               + ((-theta * log(1 - exp(-(lambda 
               * x1)^alpha)))) + ((log(theta)
               + (theta - 1) * log(1 
               - exp(-(lambda * x1)^alpha)) 
               - (lambda * x1)^alpha + log(alpha) 
               + alpha * log(lambda) 
               + (alpha - 1) * log(x1))) + (v-1) * (log((-theta 
               * log(1 - exp(-(lambda * x2)^alpha))))) 
               + ((-theta * log(1 - exp(-(lambda * x2)^alpha)))) 
               + ((log(theta) + (theta - 1) 
               * log(1 - exp(-(lambda * x2)^alpha)) 
               - (lambda * x2)^alpha 
               + log(alpha) + alpha 
               * log(lambda) + (alpha - 1) * log(x2))) + (-((-theta 
               * log(1 - exp(-(lambda * x1)^alpha)))^(v) 
               + (-theta * log(1 - exp(-(lambda 
               * x2)^alpha)))^(v))^(1/v)) 
               + (log((v-1) * ((-theta 
               * log(1 - exp(-(lambda * x1)^alpha)))^(v) 
               + (-theta * log(1-exp(-(lambda 
               * x2)^alpha)))^(v))^((1 - 2 * v) / v) 
               + ((-theta * log(1 - exp(-(lambda 
               * x1)^alpha)))^(v) 
               + (-theta * log(1 - exp(-(lambda 
               * x2)^alpha)))^(v))^((2 - 2 * v) / v))))
    dloglike <- D(loglik, "lambda")
    ddl      <- D(dloglike, "lambda")
    dlda     <- D(dloglike, "alpha")
    dldt     <- D(dloglike, "theta")
    dldv     <- D(dloglike, "v")
    

     lambda <- lambda1
     alpha  <- alpha1
     theta  <- theta1
     v      <- v1
     x1     <- x[, 1]
     x2     <- x[, 2]
     ll     <- eval(ddl)
     la     <- eval(dlda)
     lt     <- eval(dldt)
     lv     <- eval(dldv)
     ans    <- c(-1 * sum(ll), -1 * sum(la), 
                 -1 * sum(lt), -1 * sum(lv))
    return(ans)
   }
  fishtheta<- function(x, lambda1, alpha1, theta1, v1)
   {
         ## Partial Derivatives
         ## wrt theta
         loglik <- expression(
               (v - 1) * (log((-theta 
               * log(1 - exp(-(lambda * x1)^alpha))))) 
               + ((-theta * log(1 - exp(-(lambda 
               * x1)^alpha)))) + ((log(theta)
               + (theta - 1) * log(1 
               - exp(-(lambda * x1)^alpha)) 
               - (lambda * x1)^alpha + log(alpha) 
               + alpha * log(lambda) 
               + (alpha - 1) * log(x1))) + (v-1) * (log((-theta 
               * log(1 - exp(-(lambda * x2)^alpha))))) 
               + ((-theta * log(1 - exp(-(lambda 
               * x2)^alpha)))) 
               + ((log(theta) + (theta - 1) 
               * log(1 - exp(-(lambda * x2)^alpha)) 
               - (lambda * x2)^alpha 
               + log(alpha) + alpha 
               * log(lambda) + (alpha - 1) * log(x2))) + (-((-theta 
               * log(1 - exp(-(lambda * x1)^alpha)))^(v) 
               + (-theta * log(1 - exp(-(lambda 
               * x2)^alpha)))^(v))^(1/v)) 
               + (log((v-1) * ((-theta 
               * log(1 - exp(-(lambda * x1)^alpha)))^(v) 
               + (-theta * log(1-exp(-(lambda 
               * x2)^alpha)))^(v))^((1 - 2 * v) / v) 
               + ((-theta * log(1 - exp(-(lambda 
               * x1)^alpha)))^(v) 
               + (-theta * log(1 - exp(-(lambda 
               * x2)^alpha)))^(v))^((2 - 2 * v) / v))))
               
    dloglike <- D(loglik, "theta")
    dtdt     <- D(dloglike, "theta")
    dtdv     <- D(dloglike, "v")
    

     lambda <- lambda1
     alpha  <- alpha1
     theta  <- theta1
     v      <- v1
     x1     <- x[, 1]
     x2     <- x[, 2]
     tt     <- eval(dtdt)
     tv     <- eval(dtdv)
     ans    <- c(-1 * sum(tt), -1 * sum(tv))
    return(ans)
  }
 fishv<- function(x, lambda1, alpha1, theta1, v1)
  {
  ## Partial Derivatives
  ## wrt nu
         loglik <- expression(
               (v - 1) * (log((-theta 
               * log(1 - exp(-(lambda * x1)^alpha))))) 
               + ((-theta * log(1 - exp(-(lambda 
               * x1)^alpha)))) + ((log(theta)
               + (theta - 1) * log(1 
               - exp(-(lambda * x1)^alpha)) 
               - (lambda * x1)^alpha + log(alpha) 
               + alpha * log(lambda) 
               + (alpha - 1) * log(x1))) + (v-1) * (log((-theta 
               * log(1 - exp(-(lambda * x2)^alpha))))) 
               + ((-theta * log(1 - exp(-(lambda * x2)^alpha)))) 
               + ((log(theta) + (theta - 1) 
               * log(1 - exp(-(lambda * x2)^alpha)) 
               - (lambda * x2)^alpha 
               + log(alpha) + alpha 
               * log(lambda) + (alpha - 1) * log(x2))) + (-((-theta 
               * log(1 - exp(-(lambda * x1)^alpha)))^(v) 
               + (-theta * log(1 - exp(-(lambda 
               * x2)^alpha)))^(v))^(1/v)) 
               + (log((v-1) * ((-theta 
               * log(1 - exp(-(lambda * x1)^alpha)))^(v) 
               + (-theta * log(1-exp(-(lambda 
               * x2)^alpha)))^(v))^((1 - 2 * v) / v) 
               + ((-theta * log(1 - exp(-(lambda 
               * x1)^alpha)))^(v) 
               + (-theta * log(1 - exp(-(lambda 
               * x2)^alpha)))^(v))^((2 - 2 * v) / v))))
               
    dloglike <- D(loglik, "v")
    dvdv     <- D(dloglike, "v")
    

     lambda <- lambda1
     alpha  <- alpha1
     theta  <- theta1
     v      <- v1
     x1     <- x[, 1]
     x2     <- x[, 2]
     vv     <- eval(dvdv)
     ans     <- -1 * sum(vv)
    return(ans)
  }
  info[1:4, 1] <- fishlambda(x, par[1], par[2], par[3], par[4])
  info[1:4, 2] <- c(info[2,1], fishalpha(x, 
                    par[1], par[2], par[3], par[4]))
  info[1:4, 3] <- c(info[3,1], info[3,2], 
                    fishtheta(x, par[1], par[2], par[3], par[4]))
  info[1:4, 4] <- c(info[4,1],info[4,2],info[4,3],
                    fishv(x, par[1], par[2], par[3], par[4]))
  dist <- (t( par - parnot) %*% info 
           %*% (par - parnot))
  return(dist)
  }
\end{verbatim}
\subsection{Likelihood Ratio Test}
\begin{verbatim}
likeratio<-function(parnot, parmle, x)
{
 ## parnot is the null parameters
 ## parmle is the mles
 ## x is the sample
 loglik <- function(par,x)
   {
    ## defining a likelihood function 
     n <- length(x[, 1])
     lambda <- par[1]
     alpha  <- par[2]
     theta  <- par[3]
     v      <- par[4]

     if (lambda <= 0 || alpha <= 0 || theta <= 0 || v < 1)
     {
       ans <- Inf
     }
     else
     {  
       R1 <- -theta * log(1 - exp(-(lambda * x[1:n, 1])^alpha))
       R2 <- -theta * log(1 - exp(-(lambda * x[1:n,2])^alpha))
       r1 <- (log(theta) + (theta - 1) 
             * log(1 - exp(-(lambda * x[1:n, 1])^alpha)) 
             - (lambda * x[1:n, 1])^alpha + log(alpha) 
             + alpha * log(lambda) 
             + (alpha-1) * log(x[1:n, 1]))
       r2 <- (log(theta) + (theta - 1) 
              * log(1 - exp(-(lambda * x[1:n, 2])^alpha)) 
              - (lambda*x[1:n, 2])^alpha + log(alpha) 
              + alpha * log(lambda) 
              + (alpha - 1) * log(x[1:n,2]))


       s11 <- (v - 1) * sum(log(R1)) + sum(R1)
       s12 <- sum(r1)
       s21 <- (v - 1) * sum(log(R2)) + sum(R2)
       s22 <- sum(r2)
       s3  <- sum(-(R1^(v) + R2^(v))^(1 / v))
       s4  <- sum(log((v - 1) * (R1^(v) + R2^(v))^((1 - 2 * v) / v)
                  + (R1^(v) + R2^(v))^((2 - 2 * v) / v)))

       ans<- (s11+s12+s21+s22+s3+s4)
      }

     return(ans)
   }
  ratio<- -2*(loglik(parnot,x)-loglik(parmle,x))
  
  return(ratio)
}
\end{verbatim}
\subsection{Bootstrap Based Mahalonobis Distance}
\begin{verbatim}
Bootstrapdist <- function(par, parnot, x, B, iter)
 {
 bootstrap.cov <- function(par, x, B, iter)
 {
 ##Inputing copula sampler
 
 sample<-function(lambda, alpha, theta, v1, n, iter)
  {
    #Defining the copula
    f <- function(w, x, v, t) 
    { 
      copula <- (exp(-((-log(w))^v + (-log(x))^v)^(1 / v)) 
       * ((-log(w))^v + (-log(x))^v)^((1 - v) / v)
       * (-log(w))^(v - 1) * (1 / w) - t)
    }
    ## Attempting 1st sample
    u <- matrix(0,n,2)
    for (i in 1:n)
    {
     u1 <- vector(, iter)
     u2 <- vector(, iter)
     u1[1] <- runif(1, 0, 1)
     u2[1] <- runif(1, 0, 1)
     for (j in 2:iter)
     {
      t <- runif(1, 0, 1)
      u1[j] <- NA
      try(u1[j] <- uniroot(f, c(0,1), w = u2[j - 1], 
                           v = v1, t = t)$root, silent = T)
      if (is.na(u1[j]))
      {
        u1[j] <- 0
      }
      t <- runif(1, 0, 1)
      u2[j] <- NA
      try(u2[j] <- uniroot(f, c(0, 1), w = u1[j], 
                           v = v1,t = t)$root, silent=T)
      if (is.na(u2[j]))
      {
        u2[j] <- 0
      }
     }
     u[i,1] <- u1[iter]
     u[i,2] <- u2[iter]
    }
    y <- (1 / lambda) * (-log(1 - u^(1 / theta)))^(1 / alpha)
    for (i in 1:n)
    {
     ## If components are bad, resample
     while (y[i, 1] == Inf || y[i, 2] == Inf || 
            y[i,1] == 0 || y[i,2] == 0)
     {
      u1 <- vector( ,iter)
      u2 <- vector( ,iter)
      u1[1] <- runif(1, 0, 1)
      u2[1] <- runif(1, 0, 1)
      for (j in 2:iter)
      {
       t <- runif(1, 0, 1)
       u1[j] <- NA
       try(u1[j] <- uniroot(f, c(0, 1), w = u2[j - 1], 
                            v = v1, t = t)$root, silent=T)
       if (is.na(u1[j]))
       {
        u1[j] <- 0
       }
       t <- runif(1,0,1)
       try(u2[j] <- uniroot(f, c(0, 1), w = u1[j],
                            v = v1, t = t)$root, silent=T)
       if (is.na(u2[j]))
       {
        u2[j] <- 0
       }
      }
      u[i,1] <- u1[iter]
      u[i,2] <- u2[iter]
      y[i, ] <- ((1 / lambda) 
                * (-log(1 - u[i, ]^(1 / theta)))^(1 / alpha))
      }
     }
     return(y)
    }
    ##Resampling
    l3 <- matrix(0,B,1)
    a3 <- matrix(0,B,1)
    t3 <- matrix(0,B,1)
    v3 <- matrix(0,B,1)
    n  <- length(x[,1])
    for(j in 1:B)
    {
     x3    <- sample(par[1], par[2], par[3], par[4], n, iter)
     z3    <- mmle(x3, par[1], par[2], par[3], par[4])$par
     l3[j] <- z3[1]
     a3[j] <- z3[2]
     t3[j] <- z3[3]
     v3[j] <- z3[4]
    }
    boot <- cbind(l3, a3, t3, v3)
    S2b  <- cov(boot)
    return(S2b)
   }
   ## Mahalanobis Distance
   dist <- (t(par - parnot) 
             %*% solve(bootstrap.cov(par, x, B, iter),(par - parnot)))
   return(dist)
 }   
 
\end{verbatim}
\end{singlespacing}
  
\backmatter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Backmatter -- Bibliography, appendices, etc.%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%assuming that you're entering the items directly.     %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%If there's only one appendix, just call it %
%"Appendix".  Otherwise, use "Appendix A",  %
%"Appendix B", etc.                         %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\end{document}




