On Identity Tests for High Dimensional Data Using RMT

TL;DR

This paper redefines two key identity tests for high-dimensional data using random matrix theory, improving their applicability to non-Gaussian data with unknown means and demonstrating superior performance in simulations.

Contribution

The authors introduce modified versions of CLRT and LW tests that handle unknown means and non-Gaussian distributions, enhancing high-dimensional identity testing methods.

Findings

New tests perform well in size and power across various data distributions.

CLRT is more sensitive to eigenvalues less than 1.

LW test better detects eigenvalues greater than 1.

Abstract

In this work, we redefined two important statistics, the CLRT test (Bai et.al., Ann. Stat. 37 (2009) 3822-3840) and the LW test (Ledoit and Wolf, Ann. Stat. 30 (2002) 1081-1102) on identity tests for high dimensional data using random matrix theories. Compared with existing CLRT and LW tests, the new tests can accommodate data which has unknown means and non-Gaussian distributions. Simulations demonstrate that the new tests have good properties in terms of size and power. What is more, even for Gaussian data, our new tests perform favorably in comparison to existing tests. Finally, we find the CLRT is more sensitive to eigenvalues less than 1 while the LW test has more advantages in relation to detecting eigenvalues larger than 1.