Asymptotic expansions of the non-null distributions of the likelihood ratio criteria for testing the equality of a covariance matrix, equality of a mean vector and a covariance matrix, independence between two sets of variates, equality of two covariance matrices, in multivariate normal distributions are derived under the sequence of alternative hypotheses converging to the null hypothesis when the sample size tends to infinity. Numerical accuracies of the asymptotic formulas are also examined.
"Asymptotic Non-Null Distributions of the Likelihood Ratio Criteria for Covariance Matrix Under Local Alternatives." Ann. Statist. 1 (4) 718 - 728, July, 1973. https://doi.org/10.1214/aos/1176342466