  We investigate the likelihood ratio test for a large block-diagonal
covariance matrix with an increasing number of blocks under the null
hypothesis. While so far the likelihood ratio statistic has only been studied
for normal populations, we establish that its asymptotic behavior is invariant
under a much larger class of distributions. This implies robustness against
model misspecification, which is common in high-dimensional regimes.
Demonstrating the flexibility of our approach, we additionally establish
asymptotic normality of the log-likelihood ratio test statistic for the
equality of many large sample covariance matrices under model uncertainty. For
this statistic, a subtle adjustment to the centering term is needed compared to
normal case. A simulation study and an analysis of a data set from psychology
emphasize the usefulness of our findings.
