Wavelet thresholding in nonparametric regression with change-point applications

Abstract

Two research areas that have generated a great deal of interest in the field of statistics are the change-point problem and nonparametric regression. This work is concerned with a melding of these two general ideas. In this situation, the interest is in estimating regression functions that are suspected to have abrupt jumps or other unusual features. Much of the current research in nonparametric regression involves methods which rely on various assumptions on the smoothness of the underlying function, so such methods are inappropriate in this setting. The field of wavelets has received a good amount of interest in many fields of applied mathematics for its ability to handle functions with jumps or other unusual features. The application of wavelets to nonparametric regression began with Donoho and Johnstone (1992a), who introduced the soft thresholding operator. This dissertation develops two new data-dependent wavelet thresholding techniques and compares them with existing techniques. These new techniques are shown to perform fairly well in terms of mean squared error and also in terms of visual quality of the estimators. They perform well for flat functions, as well as for functions with numerous jumps. Various issues such as boundary handling, sample size considerations, and computational concerns are addressed. All the wavelet thresholding techniques are compared using simulated data sets. Finally, the new thresholding procedures are applied to two real data sets.