Some statistical methods for directly and indirecly observed functional data
In this dissertation, we will be concerned with the statistical inference regarding linear models with functional data. For the sake of generality these functional data will be considered as sample elements in an abstract infinite dimensional Hilbert space. In the special instance of the one-sample problem, both directly and indirectly observed functions will be included. It should be stressed that the linear model mentioned above each sample elements itself is a function, so that we have more information than in cases where the data consist of a number of sampled function values. In Chapter 1, we will review some useful properties and formulas of arbitrary random variables and Gaussian random variables in Hilbert spaces. It should be noted that a Gaussian measure will be employed as a dominating measure because there doesn't exist a shift invariant (i.e. Lebesgue) measure on an infinite dimensional Hilbert space. In Chapter 2, linear model in Hilbert space will be considered. We will borrow the notation from the univariate linear model and use matrices to arrive at a convenient notation for linear models in Hilbert spaces. We will show that our estimator of the function parameter has approximately a Gaussian distribution for large sample size. In Chapter 3, three special cases of the main model introduced in Chapter $2$ will be considered. First, the simplest version of the one-sample problem in Hilbert spaces will be introduced together with an application of neighborhood hypotheses. Second, the indirect-one-sample problem in Hilbert spaces will be considered. We will exploit the spectral-cut-off type regularized inverse and consider the MISE of the estimator as a means to investigate its quality. In fact, we will prove that the estimator is rate-optimal. Finally, multi-sample problem will be briefly considered along the same lines as the direct one-sample problem.