Yunzhu He

Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


Mathematical Sciences

Committee Chair

Dongsheng Wu

Committee Member

Paul Jung

Committee Member

Kyle Siegrist

Committee Member

Patrick Wang

Committee Member

Mary E. Weisskopf


Wavelets (Mathematics), Regression analysis., Nonparametric statistics.


Nonparametric regression is a fundamental tool in data analysis. Wavelet analysis has been proved to be a powerful statistical technique in the nonparametric regression. Wavelet’s appealing advantages make this approach very popular in a wide range of practical applications in many fields. In this dissertation, we consider the nonparametric regression model with correlated noise by using wavelet methods. One of the main purposes of this dissertation is to perform an extensive simulation study to some wavelet methods for the univariate nonparametric regression model with long- range and short-range dependent noise and to make comparisons to performance of the wavelet approaches. The simulation experiments are implemented on a variety of sample sizes, test functions, and wavelet filters in MATLAB. Since there is no single criterion that can adequately summarize the behavior of an estimator, various criteria are used to measure performance in finite sample situations. Insight into the performance of these estimators is obtained from computational results, graphical outputs, numerical tables and figures. To better understand the wavelet analysis and simulation study, introduction to wavelets and relative preliminaries are provided. The other main purpose of this dissertation is to explore new wavelet estimators for multi-parameter nonparametric regression model with long-range correlated noise. For simplicity and convenience of the exposition, we mainly focus on the two-parameter case. We propose some nonlinear wavelet-based estimators of two-parameter mean regression function with long memory data in L2 ([0, 1]^2) . In addition, we provide an asymptotic expansion for the mean integrated squared error (MISE) of the function estimators. We also prove that this MISE expansion works not only when the underlying mean regression function is continuously differentiable but also when the underlying mean regression function is only piecewise smooth. The main results and proofs are provided in detail.



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.