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., 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. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

搜集汇总

背景与挑战

背景概述

该数据集专注于高维数据的自适应测试方法，提出了一种基于Lq范数的U统计量，用于解决包括均值向量测试、线性模型系数测试和高维观测值独立性测试等多种全局测试问题。通过优化计算方法，显著降低了传统U统计量的计算复杂度，适用于大规模数据分析。