On the sphericity test with large-dimensional observations

TL;DR

This paper introduces corrected sphericity tests for high-dimensional data, providing new formulas for their asymptotic distributions and demonstrating their effectiveness through extensive simulations.

Contribution

It develops new asymptotic formulas for sphericity tests in large dimensions, valid for non-Gaussian data with finite fourth moments, and compares their performance with existing methods.

Findings

Corrected tests improve accuracy in large dimensions.

Asymptotic distributions are derived under general conditions.

Tests show strong performance in simulations.

Abstract

In this paper, we propose corrections to the likelihood ratio test and John's test for sphericity in large-dimensions. New formulas for the limiting parameters in the CLT for linear spectral statistics of sample covariance matrices with general fourth moments are first established. Using these formulas, we derive the asymptotic distribution of the two proposed test statistics under the null. These asymptotics are valid for general population, i.e. not necessarily Gaussian, provided a finite fourth-moment. Extensive Monte-Carlo experiments are conducted to assess the quality of these tests with a comparison to several existing methods from the literature. Moreover, we also obtain their asymptotic power functions under the alternative of a spiked population model as a specific alternative.