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 paper, we propose an approximate solution to the problem by adjusting the degrees of freedom of the F distribution. Asymptotic expansions up to the term of order N-2 for the first and second moments of the U statistic are given, where N is the total sample size minus two. A new approximate degrees of freedom and its bias correction are obtained. Finally, numerical comparison is presented by a Monte Carlo simulation.