Theory and Algorithms for Penalization, Graphical Models, and Surrogate Marker Evaluation

Abstract

In this dissertation, we study three problems: oracle inequality in high-dimensional statistics theory, graphical models, and surrogate measures in clinical trials. First, we introduce a general slow rate bound for maximum regularized likelihood estimators in Kullback-Leibler divergence. The result applies to a wide variety of models and estimators where the densities have a convex parametrization, and the regularization is definite and positively homogenous. Next, we introduce a general framework, the so-called exponential trace models, for undirected graphical models. We employ a sampling-based approximation algorithm to compute the maximum likelihood estimator. The models apply to a wide range of data, such as continuous, discrete, and different combinations of those. Finally, we review the primary frameworks of surrogate measures and propose two new ones, the population surrogacy fraction of treatment effect and time-varying F-measure. The new measures complement the existing statistical framework and apply to the HIV Prevention Trial Network 052 Study.