Date of Award

12-16-2020

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Yichuan Zhao

Second Advisor

Jing Zhang

Third Advisor

Yichen Cheng

Fourth Advisor

Jun Kong

Abstract

Empirical likelihood (EL) is a nonparametric method inspired by the usual maximum likelihood. There has been a wide range of applications to different statistical parameters since Owen (1988, 1990)’s work. While EL has many advantages over existing inference methods, it also has flaws: heavy computations, low accuracy for the small samples and high-dimensional applications, etc. In this dissertation, we investigate some EL's extensions, namely the jackknife empirical likelihood (JEL), the i.i.d. empirical likelihood (IID EL), and the weighted empirical likelihood (WEL) in constructing confidence intervals (CI) for particular parameters of interest. It contributes to significantly improving the CI by reducing substantially the extensive computation associated with the EL method, ameliorating the poor performance of EL for the small sample and heavy-tailed distributions.

We propose a new plug-in approach of JEL to reduce the computational cost in comparing two Gini indices for paired data. One of the main results of the EL is the nonparametric extension of Wilks' theorem for parametric likelihood ratios. However, this result is violated when the data is censored. To circumvent this issue for some specific parameters, we combine the EL method with the influence functions (IID EL) to construct a confidence interval for the mean residual life (MRL) function in the presence of length-bias. Further, we extend the IID EL to the two-sample mean difference, where the two samples considered are right-censored. Last, we consider the weighted empirical likelihood (WEL) for comparing two correlated areas under the ROC curves (AUC).

For the first three essays, we proved that Wilks' theorem holds: the log-likelihood ratio statistic is asymptotically chi-square distributed. And the WEL statistic has a scaled chi-squared distribution. The extensive simulations demonstrated that, for finite samples, all the proposed methods outperform the existing EL methods in coverage probability accuracy and average lengths of CI. Finally, the application to real data demonstrated that the proposed methods are of practical value.

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