Mathematics Ph.D. Dissertations

Title

Jackknife Empirical Likelihood And Change Point Problems

Date of Award

2015

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Statistics

First Advisor

Wei Ning (Advisor)

Second Advisor

Arjun Gupta (Advisor)

Third Advisor

Arthur Yeh (Other)

Fourth Advisor

John Chen (Committee Member)

Abstract

Nonparametric methods are widely used in many practical applications when certain distributional assumptions on the underlying population are questionable. One popular nonparametric method is the empirical likelihood (EL) approach. Owen (1988) showed that the most appealing property of this method is that the asymptotic distribution of the empirical likelihood ratio test statistic follows a chi-squared distribution, which is the same as the asymptotic distribution of test statistic under parametric settings. However, when this approach is used on a nonlinear statistic, such as U-statistics, it will lose its computational advantage due to the increasing difficulties in simultaneously solving a number of nonlinear equations by the method of Lagrange multipliers. Jing et al. (2009) provided an example to illustrate this challenge and proposed a simpler method called jackknife empirical likelihood (JEL) method, which combines the jackknife and the empirical likelihood approaches, and Wilks' theorem can be established by the method.

This dissertation research pursues the testing procedures based on JEL for the equality of two variances and for the equality of two mean residual lifetime functions for complete data. The asymptotical distribution under null hypothesis is also explored for these two tests, respectively. The type I error and power performance of these tests are illustrated through simulation studies. Applications of the proposed tests in data analysis and comparisons with other approaches are also studied. In addition, the change-point problems for mean and variance based on JEL and for mean residual lifetime functions of independent random variables under random censorship based on EL are investigated, individually. The testing procedure for detecting change point in mean and variance is used to detect the location of the change point in the Stock Market data and the Air Traffic data from Hsu (1979). Two real data sets in the R package ``survival": Veterans' administration lung cancer data and Stanford heart transplant data are utilized to illustrate the testing procedure for detecting change in the mean residual lifetime.

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