Mathematics Ph.D. Dissertations

Title

Statistical Analysis of Skew Normal Distribution and its Applications

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 (Committee Member)

Third Advisor

John Chen (Committee Member)

Fourth Advisor

Jane Chang (Committee Member)

Abstract

In many practical applications it has been observed that real data sets are not symmetric. They exhibit some skewness, therefore do not conform to the normal distribution, which is popular and easy to be handled. Azzalini (1985) introduced a new class of distributions named the skew normal distribution, which is mathematically tractable and includes the normal distribution as a special case with skewness parameter being zero. The skew normal distribution family is well known for modeling and analyzing skewed data. It is the distribution family that extends the normal distribution family by adding a shape parameter to regulate the skewness, which has the higher flexibility in fitting a real data where some skewness is present. In this dissertation, we will explore statistical analysis related to this distribution family.

In the first part of the dissertation, we develop a nonparametric goodness-of-fit test based on the empirical likelihood method for the skew normal distribution. The empirical likelihood was proposed by Owen (1988). It is a method which combines the reliability of the canonical nonparametric method with the flexibility and effectiveness of the likelihood approach. The statistical inference of the test statistic is derived. Simulations indicate that the proposed test can control the type I error within a given nominal level, and it has competitive power comparing to the other available tests. The test is applied to IQ scores data set and Australian Institute of Sport data set to illustrate the testing procedure.

In the second part we focus on the change point problem of the skew normal distribution. The world is filled with changes, which can lead to unnecessary losses if people are not aware of it. Thus, statisticians are faced with the problem of detecting the number of change points or jumps and their location, in many practical applications. In this part, we address this problem for the standard skew normal family. We focus on the test based on the Schwartz information criterion (SIC) to detect the position and the number of change points for the shape parameter. The likelihood ratio test and the bayesian methods as two alternative approaches will be introduced briefly. The asymptotic null distribution of the SIC test statistics is derived and the critical values for different sample sizes and nominal levels are computed for the adjustified SIC test statistic. Simulation study indicates the performance of the proposed test.

In the third part of the dissertation, we extend the methods in the second part by studying the different types of change point problem for the general skew nor mal distribution, which include: the simultaneous changes of location and scale parameters, the simultaneous change of location, scale and shape parameters. We derive the test statistic based on SIC to detect and estimate the number of possible change points. Firstly, we consider the change point problem for the simultaneous changes of location and scale parameters, assuming that the shape parameter is unknown and has to be estimated. Secondly, we explore the change point problem for simultaneous changes of location, scale and shape parameters.

The asymptotic null distribution and the corresponding adjustification for the test statistic are established. Simulations for each proposed test are conducted to indicate the performance of the test. Power comparisons with the available tests are investigated to indicate the advantage of the proposed test. Applications to real data are provided to illustrate the test procedure.

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