Document Type

Dissertation

Date of Award

12-31-2021

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Sundarraman Subramanian

Second Advisor

Sunil Kumar Dhar

Third Advisor

Satrajit Roychoudhury

Fourth Advisor

Zuofeng Shang

Fifth Advisor

Antai Wang

Abstract

Two-sample location-scale refers to a model that permits a pair of standardized random variables to have a common distribution. This means that if X1 and X2 are two random variables with means µ1 and µ2 and standard deviations ?1 and ?2, then (X1 - µ1)/?1 and (X2 - µ2)/?2 have some common unspecified standard or base distribution F0. Function-based hypothesis testing for these models refers to formal tests that would help determine whether or not two samples may have come from some location-scale family of distributions, without specifying the standard distribution F0. For uncensored data, Hall et al. (2013) proposed a test based on empirical characteristic functions (ECFs), but it can not be directly applied for censored data. Empirical likelihood with minimum distance (MD) plug-ins provides an alternative to the approach based on ECFs (Subramanian, 2020). However, when working with standardized data, it appeared feasible to set up plug-in empirical likelihood (PEL) with estimated means and standard deviations as plug-ins, which avoids MD estimation of location and scale parameters and (hence) quantile estimation. This project addresses two issues: (i) Set up a PEL founded testing procedure that uses sample means and standard deviations as the plug-ins for uncensored case, and Kaplan-Meier integral based estimators as plug-ins for censored case, (ii) Extend the ECF test to accommodate censoring. Large sample null distributions of the proposed test statistics are derived. Numerical studies are carried out to investigate the performance of the proposed methods. Real examples are also presented for both the uncensored and censored cases.

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