Adaptive L-statistics for high dimensional test problem

TL;DR

This paper develops adaptive L-statistics for high-dimensional one-sample tests, deriving their distributions, establishing asymptotic independence, and proposing a Cauchy combination test to improve robustness across different sparsity levels.

Contribution

It introduces a novel adaptive testing framework combining L-statistics with different parameters for high-dimensional data analysis.

Findings

The proposed Cauchy combination test improves detection power.

Simulation results demonstrate robustness across sparsity levels.

Real-data applications validate the method's effectiveness.

Abstract

In this study, we focus on applying L-statistics to the high-dimensional one-sample location test problem. Intuitively, an L-statistic with $k$ parameters tends to perform optimally when the sparsity level of the alternative hypothesis matches $k$. We begin by deriving the limiting distributions for both L-statistics with fixed parameters and those with diverging parameters. To ensure robustness across varying sparsity levels of alternative hypotheses, we first establish the asymptotic independence between L-statistics with fixed and diverging parameters. Building on this, we propose a Cauchy combination test that integrates L-statistics with different parameters. Both simulation results and real-data applications highlight the advantages of our proposed methods.