Some Problems in Statistical Inference Under Order Restrictions.

In this dissertation, we study three problems related to statisticalinference under order restrictions of the unknown parameters. Thefirst two deal with Cox regression models with missing data, where the baseline cumulative function satisfies a monotone restriction, and the third problem isconcerned 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 proposedmethods, and our findings.In Chapter I, we consider inference in Cox regression models withgrouped survival data where some of the covariates can be missing atrandom. We propose an inverse selection probability weightedlikelihood method for fitting the Cox model to these data. We showthat, when the probabilities that the covariates are observed arereasonably estimated, the weighted likelihood estimator withestimated probabilities can be more asymptotically efficient thanthe weighted likelihood estimator that uses the true probabilities.We did a simulation study to assess the performance of the proposedmethod and applied the method to analyze data from an HIV vaccine trial study.In Chapter II, the problem is still concerned with missingcovariates in Cox regression models, but the failure time data arecurrent status data. We establish the asymptotic results of theestimator and show that the weighted likelihood estimator withestimated weights can be more efficient than the estimator using true weights. Estimation of the asymptotic variance is alsodiscussed. A case-cohort study from an HIV vaccine trial is used todemonstrate the proposed method.In Chapter III, we suppose that there are a number of binomialrandom variables with probabilities of ``success" being ordered. Forsimplicity, we focus on the situation of two such binomial randomvariables, with probabilities $p_1, p_2$ satisfying the restrictionthat $p_1le p_2$. We first derive the (non-normal) asymptoticdistribution of the restricted MLE, and then propose inferenceprocedures based on the asymptotic distributions, and a number ofbootstrap methods, and compare them in a simulation study. We foundthat the bootstrap percentile confidence interval has goodperformance and is the best amongst those considered.