Unbiasedness of the Likelihood Ratio Tests for Equality of Several Covariance Matrices and Equality of Several Multivariate Normal Populations

Random samples of size $N_\alpha$ are taken from the $p$-variate normal populations $N_p(\mu^{(\alpha)}, \Sigma^{(\alpha)}), 1 \leqslant \alpha \leqslant k$, with $\mu^{(\alpha)}$ and $\Sigma^{(\alpha)}$ unknown. Bartlett's modification of the likelihood ratio test (LRT) for the hypothesis $H_1:\Sigma^{(1)} = \cdots = \Sigma^{(k)}$ rejects $H_1$ for large values of $|S|^n/\Pi|S^{(\alpha)}|^{n_\alpha}$, where $S = \Sigma S^{(\alpha)}, n_\alpha = N_\alpha - 1, n = \Sigma n_i$, and $S^{(\alpha)}$ is the sample covariance matrix from the $\alpha$th population. The (unmodified) LRT for the hypothesis $H_1: \mu^{(1)} = \cdots = \mu^{(k)}, \Sigma^{(1)} = \cdots = \Sigma^{(k)}$ rejects $H_2$ for large values of $|S + T|^N/\Pi|S^{(\alpha)}|^{N_\alpha}$, where $N = \Sigma N_\alpha, T = \Sigma N_\alpha(\bar{X}^{(\alpha)} - \bar{X}^{(+)})(\bar{X}^{(\alpha)} - \bar{X}^{(+)})', \bar{X}^{(\alpha)}$ is the $\alpha$th sample mean, and $\bar{X}^{(+)}$ is the grand mean. It is proved that each of these tests is unbiased against all alternatives.