In this paper, asymptotic expansions of the non-null distributions of the likelihood ratio criteria are obtained for testing the hypotheses: (a) $H_1: \Sigma = \sigma^2I, \sigma^2 > 0$, (b) $H_2: \Sigma_1 = \Sigma_2$, against alternatives which are close to the hypothesis. These expansions are of chi-square type. The first problem has been considered by Sugiura (1969) but because of the singularity at the hypothesis, his expansion will not be good for alternatives close to the hypothesis. The second problem has been considered by de Waal (1970) but the expansion given by him is invalid.
"Asymptotic Expansions of the Non-Null Distributions of Likelihood Ratio Criteria for Covariance Matrices." Ann. Statist. 2 (1) 109 - 117, January, 1974. https://doi.org/10.1214/aos/1176342617