Asymptotically Independent U-Statistics in High-Dimensional Testing

• Activity Number: 163 - Biometrics Section Byar Award Student Paper Session II
• Type: Topic Contributed
• Date/Time: Tuesday, August 4, 2020 : 10:00 AM to 11:50 AM
• Sponsor: Biometrics Section
• Author(s): Yinqiu He* and Gongjun Xu and Chong Wu and Wei Pan
• Companies: University of Michigan and University of Michigan and Florida State University and University of Minnesota
• Keywords: High-dimensional hypothesis test; U-statistics; Adaptive testing

Abstract

Many high-dimensional hypothesis tests aim to globally examine marginal or low-dimensional features of a high-dimensional joint distribution, such as testing of mean vectors, covariance matrices and regression coefficients. This paper constructs a family of U-statistics as unbiased estimators of the lp-norms of those features. We show that under the null hypothesis, the U-statistics of different finite orders are asymptotically independent and normally distributed. Moreover, they are also asymptotically independent with the maximum-type test statistic, whose limiting distribution is an extreme value distribution. Based on the asymptotic independence property, we propose an adaptive testing procedure which combines p-values computed from the U-statistics of different orders. We further establish power analysis results and show that the proposed adaptive procedure maintains high power against various alternatives.