On the Domain of Attraction of a Tracy-Widom Law with Applications to Testing Multiple Largest Roots

TL;DR

This paper investigates the asymptotic behavior of the greatest root statistic in multivariate tests, demonstrating that it converges to a Gumbel distribution under certain high-dimensional conditions, enabling improved hypothesis testing.

Contribution

It establishes the domain of attraction of the Tracy-Widom law for the greatest root statistic and shows its convergence to a Gumbel distribution for large dimensions, with broad applicability.

Findings

Gumbel distribution effectively approximates the null distribution in high dimensions.

Simulation results confirm the accuracy of the asymptotic approximation for moderate dimensions.

The theoretical results apply to various testing scenarios involving the greatest root statistic.

Abstract

The greatest root statistic arises as the test statistic in several multivariate analysis settings. Suppose there is a global null hypothesis that consists of different independent sub-null hypotheses, and suppose the greatest root statistic is used as the test statistic for each sub-null hypothesis. Such problems may arise when conducting a batch MANOVA or several batches of pairwise testing for equality of covariance matrices. Using the union-intersection testing approach and by letting the problem dimension tend to infinity faster than the number of batches, we show that the global null can be tested using a Gumbel distribution to approximate the critical values. Although the theoretical results are asymptotic, simulation studies indicate that the approximations are very good even for small to moderate dimensions. The results are general and can be applied in any setting where the greatest root statistic is used, not just for the two methods we use for illustrative purposes.