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.

本文针对一类与高维数据相关的全局检验问题，提出了一类基于Lq范数（Lq-norm）的U统计量（U-statistics）。该类检验涵盖均值向量及其空间符号检验、线性模型系数的联合检验，以及高维观测的分量独立性检验等诸多场景。在原假设（null hypothesis）下，我们在温和的矩条件与累积量（cumulant）条件下，推导得到了多组基于Lq范数的U统计量的渐近正态性与相互独立性。本文提出了一种通过p值组合两个学生化Lq基检验统计量的简单方法，并证明该方法在不同稀疏性（sparsity）的备择假设下可获得优异的检验功效。本研究是对He等人相关工作的实质性拓展——后者主要聚焦于均值与协方差检验，而我们针对广泛的核函数（kernel）类别，实现了基于Lq范数的U统计量渐近独立性的一般性处理。为缓解计算负担，我们引入了所提U统计量的一种变体：通过在求和时使用单调索引，得到了具有非对称核的U统计量；并提出动态规划（dynamic programming）方法，将完整U统计量计算所需的O(nqr)复杂度降至O(nr)，其中r为核函数的阶数。数值结果进一步证实了所提自适应检验相较于若干现有同类方法的优势。本文的补充材料可在线获取，其中包含了用于复现本研究的标准化材料说明。

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。