  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.
