Mathematics Dissertations

8-9-2022

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

Yichuan Zhao

Alexandra Smirnova

Jing Zhang

Yanqing Zhang

Abstract

Nonparametric statistical inference methods have become very popular in recent years and are the preferred method of inference among many statisticians as the data are not assumed to come from any particular statistical distribution. Empirical likelihood (EL) is one of these methods and has been popular since published by Owen (1988). While it is inspired by the usual maximum likelihood methods, it also has some flaws: heavy computations, low accuracy for the small samples and skewed data. In this dissertation, we investigate some EL's extensions, i.e., the jackknife empirical likelihood (JEL), the adjusted jackknife empirical likelihood (AJEL), the mean jackknife empirical likelihood (MJEL), the mean adjusted jackknife empirical likelihood (MAJEL),the adjusted mean jackknife empirical likelihood (AMJEL), and the transformed jackknife empirical likelihood (TJEL) in constructing confidence intervals (CI) for particular parameters of interest such as the correlation coefficient in different statistical problems. These methods increase the length of the confidence intervals for the correlation coefficient resulting in better coverage probabilities and perform better than EL methods for small sample sizes. We propose a new plug-in approach of JEL to reduce the computational cost in estimating the symmetry of various statistical distributions. One of the main results of the JEL is the nonparametric extension of Wilks' theorem for parametric likelihood ratios. We explore JEL, AJEL, MJEL, AMJEL, and MAJEL to construct a confidence interval for the correlation coefficient for data with multiplicative distortion errors. Further, we explore JEL, AJEL, MJEL, and TJEL for the k-th correlation coefficient in estimating measures of symmetry for data with multiplicative distortion errors. Finally, using exponential calibration, we develop JEL methods for the correlation coefficient between two variables with additive distortion measurement errors.

DOI

https://doi.org/10.57709/30529173