  In this paper, the $k$ sample Behrens-Fisher problem is investigated in high
dimensional setting. We propose a new test statistic and demonstrate that the
proposed test is expected to have more powers than some existing test
especially when sample sizes are unbalanced. We provide theoretical
investigation as well as numerical studies on both sizes and powers of the
proposed tests and existing test. Both theoretical comparison of the asymptotic
power functions and numerical studies show that the proposed test tends to have
more powers than existing test in many cases of unbalanced sample sizes.
