On the k-sample Behrens-Fisher problem for high-dimensional data

For several decades, much attention has been paid to the two-sample Behrens-Fisher (BF) problem which tests the equality of the means or mean vectors of two normal populations with unequal variance/covariance structures. Little work, however, has been done for the k-sample BF problem for high dimensional data which tests the equality of the mean vectors of several high-dimensional normal populations with unequal covariance structures. In this paper we study this challenging problem via extending the famous Scheffe’s transformation method, which reduces the k-sample BF problem to a one-sample problem. The induced one-sample problem can be easily tested by the classical Hotelling’s T2 test when the size of the resulting sample is very large relative to its dimensionality. For high dimensional data, however, the dimensionality of the resulting sample is often very large, and even much larger than its sample size, which makes the classical Hotelling’s T2 test not powerful or not even well defined. To overcome this difficulty, we propose and study an L2-norm based test. The asymptotic powers of the proposed L2-norm based test and Hotelling’s T2 test are derived and theoretically compared. Methods for implementing the L2-norm based test are described. Simulation studies are conducted to compare the L2-norm based test and Hotelling’s T2 test when the latter can be well defined, and to compare the proposed implementation methods for the L2-norm based test otherwise. The methodologies are motivated and illustrated by a real data example.