A Two Sample Test for Mean Vectors with Unequal Covariance Matrices

In this paper, we consider testing the equality of two mean vectors with unequal covariance matrices. In the case of equal covariance matrices, we can use Hotelling’s T 2 statistic, which follows the F distribution under the null hypothesis. Meanwhile, in the case of unequal covariance matrices, the T 2 type test statistic does not follow the F distribution, and it is also difficult to derive the exact distribution. In this study, we propose an approximate solution to the problem by adjusting the degrees of freedom of the F distribution. That is, we derive an extension of the results derived by Yanagihara and Yuan (2005). Asymptotic expansions up to the term of order N−2 for the first and second moments of the test statistic are given, where N is the total sample size minus two, and a new result of the approximate degrees of freedom is obtained. Finally, numerical comparison is presented by a Monte Carlo simulation.