Adaptive Sphericity Tests for High Dimensional Data

TL;DR

This paper introduces new max-type and combined Cauchy test procedures for high-dimensional sphericity testing, outperforming traditional sum-type tests especially under sparse alternatives, as validated by simulation studies.

Contribution

It proposes novel max-type and Cauchy combination tests that improve sphericity testing accuracy in high-dimensional, sparse scenarios.

Findings

Max-type tests outperform sum-type tests under sparse alternatives.

Cauchy combination tests effectively integrate multiple test procedures.

Simulation results show superior performance of proposed methods.

Abstract

In this paper, we investigate sphericity testing in high-dimensional settings, where existing methods primarily rely on sum-type test procedures that often underperform under sparse alternatives. To address this limitation, we propose two max-type test procedures utilizing the sample covariance matrix and the sample spatial-sign covariance matrix, respectively. Furthermore, we introduce two Cauchy combination test procedures that integrate both sum-type and max-type tests, demonstrating their superiority across a wide range of sparsity levels in the alternative hypothesis. Our simulation studies corroborate these findings, highlighting the enhanced performance of our proposed methodologies in high-dimensional sphericity testi