Theses and Dissertations

DOI

10.17077/etd.3f85-41hk

Dissertation

Spring 2019

Access Restrictions

Access restricted until 07/29/2021

Degree Name

PhD (Doctor of Philosophy)

Statistics

Zimmerman, Dale

Breheny, Patrick

Chan, Kung-Sik

Huang, Yuan

Tan, Aixin

Abstract

The skewed normal (SN) distribution introduced by Azzalini has opened a new era for analyzing skewed data. The idea behind it is that it incorporates a new parameter regulating shape and skewness on the symmetric Gaussian distribution. This idea was soon extended to other symmetric distributions such as the Student's t distribution, resulting in the invention of the skew t (ST) distribution.

The multivariate versions of the two distributions, i.e. the multivariate skew normal (MSN) and multivariate skew t (MST) distributions, have received considerable attention because of their ability to t skewed data, together with some other properties such as mathematical tractability. While many researchers focus on tting the MSN and MST dis- tributions to data, in this thesis we address another important aspect of statistical modeling using those two distributions, i.e. skewness selection and estimation. Skewness selection, as we discuss it here, means identifying which components of the skewness parameter in the MSN and MST distributions are zero.

In this thesis, we begin by reviewing some important properties of the two distributions and then we describe the obstacles that block us from doing skewness selection in the direct parameterizations of the two distributions. Then, to circumvent those obstacles, we intro- duce a new parameterization to use for skewness selection. The nice properties of this new parameterization are also summarized.

After introduction of the new parameterization, we discuss our proposed methods to reach the goal of skewness selection. Particularly, we consider adding appropriate penalties to the loss functions of the MSN and MST distributions, represented in the new parameterization of the two distributions. Technical details such as initial value selection and tuning parameter selection are also discussed. Asymptotic consistency and oracle property of some of our methods are constructed.

In the later part of the thesis, we include results from some simulation studies in order to assess the performance of our proposed methods. Also, we apply our methods to three data sets. Lastly, some drawbacks and potential future work are discussed.

x, 99 pages

Bibliography

Includes bibliographical references (pages 95-99).