2021/01/29 by Yinqiu He, Gongjun Xu, Chong Wu +1 · 3 citations
Mathematics · #Random Matrices and Applications #Statistical Methods and Inference #Statistical Methods and Bayesian Inference
paper · doi:10.1214/20-aos1951
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.