On the domain of attraction of a Tracy-Widom law with applications to testing multiple largest roots

The greatest root statistic arises as a test statistic in several multivariate analysis settings. Suppose there is a global null hypothesis H0 that consists of m different independent sub null hypotheses, i.e., H0=H01∩⋯∩H0m, 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 p→∞ faster than m→∞, we show that H0 can be tested using a Gumbel distribution to approximate the critical values. Although the theoretical results are asymptotic, simulation studies indicate that the approximation is accurate 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 discussed for illustrative purposes.