Logarithmic Quantile Estimation and Its Applications to Nonparametric Factorial Designs
Open Access
- Author:
- Tabacu, Lucia Maria
- Graduate Program:
- Statistics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 12, 2014
- Committee Members:
- Manfred Heinz Denker, Dissertation Advisor/Co-Advisor
Michael G Akritas, Committee Chair/Co-Chair
John Fricks, Committee Member
Guodong Pang, Special Member - Keywords:
- logarithmic quantile
almost sure weak convergence
linear rank statistics
nonparametric factorial designs
Kruskal-Wallis test - Abstract:
- In this dissertation we prove an almost sure weak limit theorem for simple linear rank statistics for samples with continuous distribution functions. As a corollary, the result extends to samples with ties and the vector version of an almost sure central limit theorem for vectors of linear rank statistics. Moreover, we derive a weak convergence result for some quadratic forms. These results are then applied to quantile estimation and to hypothesis testing for nonparametric statistical designs, here demonstrated by the c-sample problem, where the samples may be dependent. In general, the method is known to be comparable to the bootstrap and other nonparametric methods (Thangavelu \cite{THA}, and Fridline \cite{FRI}) and we confirm this finding for the c-sample problem. This dissertation also contains a study of longitudinal data originally analyzed by Lumley \cite{LUM} using odds ratio and later by Brunner, Domhof, and Langer \cite{BR7} using rank statistics. The quantile estimation procedure developed in this dissertation is well adapted to this situation and shows similar results as in Brunner et al. \cite{BR7} with the advantage of minimal assumptions on the distributions. In order to develop the theory necessary to derive these new methods, we further develop Thangavelu's first ideas in \cite{THA} and set the framework for a decision theory based on the almost sure convergence property. Although this method seems to be similar to bootstrap it goes beyond these ideas because it is based on almost sure behavior and not on distributional behavior. The same novelty occurs when stating and proving the results on rank statistics and their quadratic forms; some ideas are similar to the distributional approach in Brunner and Denker \cite{BR1} and Brunner et al. in \cite{BR7}, but the essential new idea is to deal with almost sure terms of approximating statistics, similar to the well known Slutsky result in the distributional theory, using results by Lifshits \cite{LI1} and others.