ASYMPTOTICALLY INDEPENDENT U-STATISTICS IN HIGH-DIMENSIONAL TESTING

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 l(p)-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.

Automated summary

This paper introduces a method of using U-statistics for high-dimensional hypothesis tests, showing their asymptotic independence and normal distribution under the null hypothesis. Based on this property, an adaptive testing procedure is proposed, with high power against various alternatives.