Local Orthogonal Polynomial Expansion and Empirical Saddlepoint Approximation for Density Estimation.

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2014-11-25

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Abstract

This thesis proposes a new non-parametric density estimation method and studies further the so-called Empirical Saddlepoint Approximation (ESPA). We introduce Local Orthogonal Polynomial Expansion (LOrPE), a new method to estimate the density function of a uni-variate continuous random variable. LOrPE is related to several existing methods. In a manner of construction, it is similar to the Local Likelihood Density Estimation (LLDE). LLDE matches localized sample moments to localized population moments using the log-polynomial density approximation while LOrPE matches localized expectation values of orthogonal polynomials to their sample values using polynomial density approximation. We demonstrate that, in the limit of large bandwidth, LOrPE is equivalent to Orthogonal Series Density Estimation (OSDE). In the limit of small bandwidth, LOrPE essentially functions as Kernel Density Estimation (KDE) without boundary bias. We compare the performance of LOrPE to KDE, LLDE and OSDE in a number of tests. In terms of Mean Integrated Squared Error (MISE), our results show that with a proper balance of the two tuning parameters, bandwidth(h) and degree(M), LOrPE performs better than the other methods when reconstructing densities with sharp boundaries.

The Empirical Saddle-point Approximation (ESPA) is a potentially attractive tool for density estimation since it is entirely non-parametric and free of tuning parameters. it consists of inverting the empirical moment generating function via the saddlepoint method. We derive the first order asymptotic regime of the ESPA using both in influence functions and M-estimation methodology. These results are used in simulated data sets to construct confidence bands for the resulting densities.

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Keywords

Local orthogonal polynomial expansions, Kernel density estimation, Non-parametric density estimation, Boundary bias, Influence functions, M-estimators

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