Mathematics Ph.D. Dissertations

Title

A Study of non-central Skew t Distributions and their Applications in Data Analysis and Change Point Detection

Date of Award

2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Statistics

First Advisor

Wei Ning (Advisor)

Second Advisor

Arjun Gupta (Advisor)

Third Advisor

Junfeng Shang (Committee Member)

Fourth Advisor

Mark Earely (Committee Member)

Abstract

Over the past three decades there has been a growing interest in searching for distribution families that are suitable to analyze skewed data with excess kurtosis. The search started by numerous papers on the skew normal distribution. Multivariate t distributions started to catch attention shortly after the development of the multivariate skew normal distribution.

Many researchers proposed alternative methods to generalize the uni-variate t distribution to the multivariate case. Recently, skew t distribution started to become popular in research. Skew t distributions provide more flexibility and better ability to accommodate long-tailed data than skew normal distributions.

In this dissertation, a new non-central skew t distribution is studied and its theoretical properties are explored. Applications of the proposed non-central skew t distribution in data analysis and model comparisons are studied. An extension of our distribution to the multivariate case is presented and properties of the multivariate non-central skew t distribution are discussed. We also discuss the distribution of quadratic forms of the non-central skew t distribution. In the last chapter, the change point problem of the non-central skew t distribution is discussed under different settings. An information based approach is applied to detect the location of the change point in the non-central skew t distribution. The power of this approach is illustrated via simulation studies. Finally, the change point approach is used to detect the location of the change point in the weekly return rates of three Latin American countries using the non-central skew t distribution.

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