Date of Award

4-2016

Degree Name

Doctor of Philosophy

Department

Statistics

First Advisor

Dr. Joseph W. McKean

Second Advisor

Dr. Paul Rajib

Third Advisor

Dr. Je ffrey Terpstra

Fourth Advisor

Dr. Thomas H. Oliphant

Keywords

Bayesian, rank-based, Bayesian HBR, Linear models, generalized linear models, BRBM GLM

Abstract

A Bayesian Rank Based Method for linear models is developed in this research. The estimation of the regression coefficients is based on the full conditional distributions utilizing a rank based initial fit. The data likelihood is based on the asymptotic distribution of the gradient function and the asymptotic linearity of this rank-based procedure. Prior distributions are put on regression coefficient(s) and scale parameter(s). The effects of different priors on this scale parameter(s) are studied. Using these full conditional distributions, the estimates are obtained by a Markov Chain Monte-Carlo (MCMC) procedure. The results of our simulation studies show that these Bayesian rank-based estimates retain the robustness of the rank-based fit. In many of the situations they were more efficacious than the rank-based fits. For error distributions with heavy tails, they were much more efficient than traditional least squares estimates. These investigations also showed that incorporating prior information on the scale parameter resulted in much more efficient estimates over all situations investigated.

Although robust in the response space, similar with the rank-based initial fit, our procedure is not robust in factor space. To counter this, we also developed a Bayesian High Breakdown rank-based estimate which is robust in both the response and factor spaces.

We extended this procedure to a Generalized Linear Model (GLM) where our pseudo-likelihood is a rank based quantity that utilizes Pearson residuals. Our method is robust to outliers in the response-space for link functions that are monotone and three times continuously differentiable. As in the linear model case, our procedure incorporates the asymptotic distribution and the asymptotic linearity of the initial rank-based fit. The estimates are obtained by a MCMC procedure. An extensive simulation study based on the posterior distribution via MCMC method is performed. As in the linear model case, the Bayesian rank-based procedure is more efficient than the rank-based initial fit estimates. Also, for contaminated situations they are much more efficient than traditional maximum likelihood estimates.

Access Setting

Dissertation-Campus Only

Restricted to Campus until

4-15-2026

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