Adaptive Testing for High-Dimensional Data

In this article, we propose a class of Lq-norm based U-statistics for a family of global testing problems related to high-dimensional data. This includes testing of mean vector and its spatial sign, simultaneous testing of linear model coefficients, and testing of component-wise independence for high-dimensional observations, among others. Under the null hypothesis, we derive asymptotic normality and independence between Lq-norm based U-statistics for several qs under mild moment and cumulant conditions. A simple combination of two studentized Lq-based test statistics via their p-values is proposed and is shown to attain great power against alternatives of different sparsity. Our work is a substantial extension of He et al. (2021), which is mostly focused on mean and covariance testing, and we manage to provide a general treatment of asymptotic independence of Lq-norm based U-statistics for a wide class of kernels. To alleviate the computation burden, we introduce a variant of the proposed U-statistics by using the monotone indices in the summation, resulting in a U-statistic with asymmetric kernel. A dynamic programming method is introduced to reduce the computational cost from O(nqr), which is required for the calculation of the full U-statistic, to O(nr) where r is the order of the kernel. Numerical results further corroborate the advantage of the proposed adaptive test as compared to some existing competitors.