Different-based methods in nonparametric regression models

Abstract

This thesis develops some new di.erence-based methods for nonparametric regression models. The .rst part of this thesis focuses on the variance estimation for nonparametric models with various settings. In Chapter 2, a uni.ed framework of variance estimator is proposed for a model with smooth mean function. This framework combines the higher order di.erence sequence with least squares method and greatly extends the literature, including most of existing methods as special cases. We derive the asymp­totic mean squared errors and make both theoretical and numerical comparison for various estimators within the system. Based on the dramatic interaction of ordinary di.erence sequences and least squares method, we eventually .nd a uniformly sat­isfactory estimator for all the settings, solving the challenging problem of sequence selection. In Chapter 3, three methods are developed for the variance estimation in the repeated measurement setting. Both their asymptotic properties and .nite sample performance are explored. The sequencing method is shown to be the most adaptive while the sample variance method and the partitioning method are shown to outperform in certain cases. In Chapter 4, we propose a pairwise regression method for estimating the residual variance. Speci.cally, we regress the squared di.erence between observations on the squared distance between design points, and then es­timate the residual variance as the intercept. Unlike most existing di.erence-based estimators that require a smooth regression function, our method applies to regres­sion models with jump discontinuities. And it also applies to the situations where the design points are unequally spaced. The smoothness assumption of the nonparametric regression function is quite critical for the curve .tting and the residual variance estimation. The second part (Chapter 5) concentrates on the discontinuities detection for the mean function. In particular, we revisit the di.erence-based method in M¨uller and Stadtm¨uller (1999) and propose to improve it. To achieve the goal, we .rst reveal that their method is less e.cient due to the inappropriate choice of the response variable in their linear regression model. We then propose a new regression model for estimating the resid­ual variance and the total amount of discontinuities simultaneously. In both theory and simulations, we show that the proposed variance estimator has a smaller MSE compared to their estimator, whereas the e.ciency of the estimators for the total amount of discontinuities remain unchanged. Finally, we construct a new test proce­dure for detection using the newly proposed estimations; and via simulation studies, we demonstrate that our new test procedure outperforms the existing one in most settings. At the beginning of Chapter 6, a series of new di.erence sequences is de.ned to complete the span between the optimal sequence and the ordinary sequence. The vari­ance estimators using proposed sequences are shown to be quite robust and achieve smallest mean square errors for most of general settings. Then, the di.erence-based methods for variance function estimation are generally discussed. Keywords: Asymptotic normality, Di.erence-based estimator, Di.erence sequence, Jump point, Least square, Nonparametric regression, Pairwise regression, Repeated measurement, Residual variance

Date of Award	31 Jul 2014
Original language	English
Supervisor	Tiejun TONG (Supervisor)