Some Interactions of Modern Optimization and Statistics

Abstract

This dissertation attacks several challenging problems using state-of-the-art modern optimization and statistics. We first consider optimal, two stage, adaptive enrichment designs for randomized trials, using sparse linear programming. Adaptive enrichment designs involve preplanned rules for modifying enrollment criteria based on accruing data in a randomized trial. Such designs have been proposed. The goal is to learn which populations benefit from an experimental treatment. Two critical components of adaptive enrichment designs are the decision rule for modifying enrollment, and the multiple testing procedure. We provide the first general framework for simultaneously optimizing both of these components for two stage, adaptive enrichment designs. We minimize expected sample size under constraints on power and the familywise Type I error rate.

Next, we consider high-dimensional spatial graphical model estimation under a total cardinality constraint. Though this problem is highly nonconvex, we show that its primal-dual gap diminishes linearly with the dimensionality and provide a convex geometry justification of this `blessing of massive scale' phenomenon. Motivated by this result, we propose an efficient algorithm to solve the dual problem and prove that the solution achieves optimal statistical properties.

Finally, we consider the problem of hypothesis testing and confidence intervals under high dimensional proportional hazards models. Motivated by a geometric projection principle, we propose a unified likelihood ratio inferential framework, including score, Wald and partial likelihood ratio statistics for hypothesis testing. Without assuming model selection consistency, we derive the asymptotic distributions of these test statistics, establish their semiparametric optimality, and conduct power analysis under Pitman alternatives. We also develop procedures to construct pointwise confidence intervals for the baseline hazard function and conditional hazard function.