Some Problems in Statistical Inference Under Order Restrictions.

Abstract

In this dissertation, we study three problems related to statistical inference under order restrictions of the unknown parameters. The first two deal with Cox regression models with missing data, where the baseline cumulative function satisfies a monotone restriction, and the third problem is concerned with inference of ordered probabilities in binomial random variables. In the preface, we give an introductory description of the topics we will deal with, including motivation, our proposed methods, and our findings.

In Chapter I, we consider inference in Cox regression models with grouped survival data where some of the covariates can be missing at random. We propose an inverse selection probability weighted likelihood method for fitting the Cox model to these data. We show that, when the probabilities that the covariates are observed are reasonably estimated, the weighted likelihood estimator with estimated probabilities can be more asymptotically efficient than the weighted likelihood estimator that uses the true probabilities. We did a simulation study to assess the performance of the proposed method and applied the method to analyze data from an HIV vaccine trial study.

In Chapter II, the problem is still concerned with missing covariates in Cox regression models, but the failure time data are current status data. We establish the asymptotic results of the estimator and show that the weighted likelihood estimator with estimated weights can be more efficient than the estimator using true weights. Estimation of the asymptotic variance is also discussed. A case-cohort study from an HIV vaccine trial is used to demonstrate the proposed method.

In Chapter III, we suppose that there are a number of binomial random variables with probabilities of ``success" being ordered. For simplicity, we focus on the situation of two such binomial random variables, with probabilities $p_1, p_2$ satisfying the restriction that $p_1le p_2$. We first derive the (non-normal) asymptotic distribution of the restricted MLE, and then propose inference procedures based on the asymptotic distributions, and a number of bootstrap methods, and compare them in a simulation study. We found that the bootstrap percentile confidence interval has good performance and is the best amongst those considered.