Approximation of Quantiles of Rank Test Statistics Using Almost Sure Limit Theorems

<p>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.</p> <p>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.</p> <p>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.</p> <p>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.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics & Statistics

Year dc:date.available

2018

Author dc:creator

• Ledbetter, Mark

Contributors dc:contributor

• Lucia Tabacu
• N. Rao Chaganty
• Norou Diawara
• Juan Du
• Manfred Denker

• Almost sure central limit theorem
• Change-point problem
• Logarithmic quantile estimation
• Nonparametric statistics
• Patterned alternatives
• Two-sample problem
• Statistics and Probability

Statement dc:rights

• <p>In Copyright. URI: <a href="http://rightsstatements.org/vocab/InC/1.0/">http://rightsstatements.org/vocab/InC/1.0/</a> This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).</p>

Identifier

9780438991675

OAI identifier oai:identifier

oai:digitalcommons.odu.edu:mathstat_etds-1005