Smooth regression quantile estimation

Yu, Keming (1997). Smooth regression quantile estimation. PhD thesis The Open University.

Abstract

In this thesis, attention will be mainly focused on the local linear kernel regression quantile estimation. Different estimators within this class have been proposed, developed asymptotically and applied to real applications. I include algorithmdesign and selection of smoothing parameters.

Chapter 2 studies two estimators, first a single-kernel estimator based on "check function" and a bandwidth selection rule is proposed based on the asymptotic MSE of this estimator. Second a recursive double-kernel estimator which extends Fan et al's (1996) density estimator, and two algorithms are given for bandwidth selection.

In Chapter 3, a comparison is carried out of local constant fitting and local linear fitting using MSEs of the estimates as a criterion.

Chapter 4 gives a theoretical summary and a simulation study of local linear kernel estimation of conditional distribution function. This has a special interest in itself as well as being related to regression quantiles.

In Chapter 5, a kernel-version method of LMS (Cole and Green, 1992) is considered. The method proposed, which is still a semi-parametric one, is based on a general idea of local linear kernel approach of log-likelihood model.

Chapter 6 proposes a two-step method of smoothing regression quantiles called BPK. The method considered is based on the idea of combining k- NN method with Healy's et al (1988) partition rule, and correlated regression model are involved.

In Chapter 7, methods of regression quantile estimation are compared for different underlying models and design densities in a simulation study. The ISE criterion of interior and boundary points is used as a basis for these comparisons. Three methods are recommended for quantile regression in practice, and they are double kernel method, LMS method and Box partition kernel method (BPK).

In Chapter 8, attention is turned to a novel idea of local polynomial roughness penalty regression model, where a purely theoretical framework is considered.

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