# Adaptive Testing for High-dimensional Data

arXiv: https://arxiv.org/abs/2303.08197 · HTML: https://mlsanity.com/paper/2303.08197/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Mar 2023.

## Built on by (most cited first, 1 of 1)

- [On computing and the complexity of computing higher-order U-statistics, exactly](https://mlsanity.com/paper/2508.12627/) (2025)

## Abstract (arXiv)

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 studies further corroborate the advantage of the proposed adaptive test as compared to some existing competitors.