Minimax Hypothesis Testing in Large-Scale Inference

This thesis studies some hypothesis testing, and related functional estimation, problems in models motivated by large-scale inference applications. Specifically, problems in sparse mixture detection, null distribution estimation, global null testing under correlation, variance estimation in compound decision theory, and testing with heteroskedastic counts are addressed. The perspective is minimax, and the goal is to characterize either the optimal functional estimation rate or the optimal testing rate in the style of Ingster. A recurring motif in the results is the emergence of rate phenomena and phase transitions distinct from the corresponding theory of estimating the underlying, high-dimensional parameter.