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

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