#### Date of Award

Winter 2018

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### Committee Director

Lucia Tabacu

#### Committee Member

N. Rao Chaganty

#### Committee Member

Norou Diawara

#### Committee Member

Juan Du

#### Committee Member

Manfred Denker

#### Abstract

There are many problems in statistics where the analysis is based on asymptotic distributions. In some cases, the asymptotic distribution is in an open form or is intractable. One possible solution is the logarithmic quantile estimation (LQE) method introduced by Thangavelu (2005) for rank tests and Fridline (2010) for the correlation coefficient. LQE is derived from an almost sure version of the central limit theorem using the results of Berkes and Csaki (2001), and it estimates the quantiles of a test statistic using only the data. To date, LQE has been used in only a few applications. We extend the use of LQE to three widely analyzed problems.

We investigate the LQE approach using fully nonparametric rank statistics to test for known trend and umbrella patterns in the main effects of three widely used factorial designs: a two-factor fixed effect model, a partial hierarchical repeated measures mixed effect model, and a mixed effect cross-classification repeated measures model. We also test for patterned alternatives in the interaction between the main effect and time in the partial hierarchical repeated measures model. We derive the almost sure central limit theorems for all of these problems and determine the level and power.

The Pettitt (1979) test is a nonparametric test based on the Mann-Whitney statistic used to detect a change in distribution in a sequence of random variables. The proposed statistic has an asymptotic distribution that is the distribution of the supremum of the absolute value of the Brownian bridge, which has an open form. We propose an approximation of the quantiles for the test statistic based on LQE. We provide simulation results for Type I error and power of the logarithmic quantile estimates for the test statistic, and compare the LQE results with other methods for two real data examples.

Thangavelu (2005) considered LQE for the nonparametric Behrens-Fisher problem with some success by introducing new numerically determined coefficients. We examine the nonparametric two-sample problem using an empirical process of U-statistic structure (Denker and Puri, 1992). Specifically, we investigate using LQE with a second order U-statistic for paired averages within each sample. We provide simulation results to show almost sure convergence of the new test statistic.

#### DOI

10.25777/64tc-0n41

#### ISBN

9780438991675

#### Recommended Citation

Ledbetter, Mark.
"Approximation of Quantiles of Rank Test Statistics Using Almost Sure Limit Theorems"
(2018). Doctor of Philosophy (PhD), Dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/64tc-0n41

https://digitalcommons.odu.edu/mathstat_etds/5

#### ORCID

0000-0001-9824-4668