In this article, we propose a class of $L_q$-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 $L_q$-norm based U-statistics for several $q$s under mild moment and cumulant conditions. A simple combination of two studentized $L_q$-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 $L_q$-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 $O(n^{qr})$, which is required for the calculation of the full U-statistic, to $O(n^r)$ where $r$ is the order of the kernel. Numerical studies further corroborate the advantage of the proposed adaptive test as compared to some existing competitors.
翻译:在本篇文章中,我们为一组与高维数据有关的全球测试问题提出了一类以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以学生为单位、以学生为单位、以美元为单位、以空间标志为单位、同时测试线性模型系数系数,并同时测试高维度观测等。根据无效假设,我们为若干以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位、以美元为单位的货币为单位的货币为单位的货币计算计算计算计算计算计算计算计算计算方法的计算方法,我们的工作是大量计算方法的计算计算方法计算方法计算方法计算。</s>