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Title: Asymptotically independent U-statistics for high dimensional adaptive testing Authors:  Gongjun Xu - University of Michigan (United States) [presenting]
Abstract: Many high dimensional hypothesis tests examine the moments of the distributions that are of interest, such as testing of mean vectors and covariance matrices. We propose a general framework that constructs a family of U statistics as unbiased estimators of those moments. The usage of the framework is illustrated by testing off-diagonal elements of a covariance matrix. We show that under null hypothesis, the U statistics of different finite orders are asymptotically independent and normally distributed. Moreover, they are also asymptotically independent with the max-type test statistic. Based on the asymptotic independence property, we construct an adaptive testing procedure that maintains high power across a wide range of alternatives. Simulation and real data are further used to validate the proposed method.