Asymptotic Independent U-Statistics in High-Dimensional Adaptive Testing

• Activity Number: 170 - Theory and Methods for High-Dimensional Data
• Type: Contributed
• Date/Time: Monday, July 30, 2018 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Yinqiu He* and Gongjun Xu and Chong Wu and Wei Pan
• Companies: University of Michigan and University of Michigan and University of Minnesota and University of Minnesota
• Keywords: high dimension; adaptive testing

Abstract

Many high dimensional hypothesis testings examine the moments of the distributions that are of interest, such as testing of mean vectors and covariance matrices. We propose a framework that constructs a family of U statistics as unbiased estimators of those moments. In this talk, the usage of the framework is illustrated by testing for independence. We show that under null hypothesis, when both data dimension and sample size go to infinity, U statistics of different finite orders are asymptotically independent and normally distributed. Moreover, they are also asymptotically independent of the max-type test statistic, whose limiting distribution is an extreme value distribution. Based on the asymptotic independence property, we construct an adaptive testing procedure which combines p values computed from U statistics of different orders. Since higher order U statistics are usually more powerful against sparse alternatives and lower order U statistics are usually more powerful against dense alternatives, this adaptive procedure is powerful against different alternatives.