High Dimensional Rank Tests for Sphericity

TL;DR

This paper introduces high-dimensional rank-based tests for sphericity that are robust, efficient, and do not require nuisance parameter estimation, leveraging the benefits of high dimensionality.

Contribution

It proposes Spearman's rho-type and Kendall's tau-type sphericity tests that are equivalent and do not need nuisance parameter estimation in high dimensions.

Findings

Tests are asymptotically normal under elliptical distributions.

Simulations show robustness and efficiency across various settings.

No need to estimate nuisance parameters in high dimensions.

Abstract

Sphericity test plays a key role in many statistical problems. We propose Spearman's rho-type rank test and Kendall's tau-type rank test for sphericity in the high dimensional settings. We show that these two tests are equivalent. Thanks to the "blessing of dimension", we do not need to estimate any nuisance parameters. Without estimating the location parameter, we can allow the dimension to be arbitrary large. Asymptotic normality of these two tests are also established under elliptical distributions. Simulations demonstrate that they are very robust and efficient in a wide range of settings.