Degree Type

Dissertation

Date of Award

2018

Degree Name

Doctor of Philosophy

Department

Statistics

Major

Statistics

First Advisor

Zhengyuan Zhu

Second Advisor

Vivekananda Roy

Abstract

In this thesis, two topics are studied, generalized linear mixed models and spatial subgroup analysis.

Within the topic of generalized linear mixed models, this thesis focuses on three aspects. First, estimation of link function in generalized linear models is studied. We propose a new algorithm that uses P-spline for nonparametrically estimating the link function which is guaranteed to be mono- tone. We also conduct extensive simulation studies to compare our nonparametric approach with various parametric approaches. Second, a spatial hierarchical model based on generalized Dirichlet distribution is developed to construct small area estimators of compositional proportions in the National Resources Inventory survey. At the observation level, the standard design based estima- tors of the proportions are assumed to follow the generalized Dirichlet distribution. After proper transformation of the design based estimators, beta regression is applicable. We consider a logit mixed model for the expectation of the beta distribution, which incorporates covariates through fixed effects and spatial effect through a conditionally autoregressive process. Finally, convergence rates of Markov chain Monte Carlo algorithms for Bayesian generalized linear mixed models are studied. For Bayesian probit linear mixed models, we construct two-block Gibbs samplers using the data augmentation (DA) techniques and prove the geometric ergodicity of the Gibbs samplers under both proper priors and improper priors. We also provide conditions for posterior propriety when the design matrices take commonly observed forms. For Bayesian logistic regression models, we establish that the Markov chain underlying Polson et al.’s (2013) DA algorithm is geometri- cally ergodic under a flat prior. For Bayesian logistic linear mixed models, we construct a two-block Gibbs sampler using Polson et al.’s (2013) DA technique under proper priors and prove the uniform ergodicity of this Gibbs sampler.

The other topic is spatial subgroup analysis with repeated measures. We use pairwise concave penalties for the differences among group regression coefficients based on smoothly clipped absolute deviation penalty. We also consider pairwise weights associated with each paired penalty based on spatial information. We show that the oracle estimator based on weighted least square is a local minimizer of the objective function with probability approaching 1 under some conditions. In the simulation study, we compare the performances of different weights as well as equal weights, which shows that the spatial information will help when the minimal group difference is small or the number of repeated measures is small.

Copyright Owner

Xin Wang

Language

en

File Format

application/pdf

File Size

171 pages

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