  The problem of testing the equality of mean vectors for high-dimensional data
has been intensively investigated in the literature. However, most of the
existing tests impose strong assumptions on the underlying group covariance
matrices which may not be satisfied or hardly be checked in practice. In this
article, an F-type test for two-sample Behrens--Fisher problems for
high-dimensional data is proposed and studied. When the two samples are
normally distributed and when the null hypothesis is valid, the proposed F-type
test statistic is shown to be an F-type mixture, a ratio of two independent
chi-square-type mixtures. Under some regularity conditions and the null
hypothesis, it is shown that the proposed F-type test statistic and the above
F-type mixture have the same normal and non-normal limits. It is then justified
to approximate the null distribution of the proposed F-type test statistic by
that of the F-type mixture, resulting in the so-called normal reference F-type
test. Since the F-type mixture is a ratio of two independent chi-square-type
mixtures, we employ the Welch--Satterthwaite chi-square-approximation to the
distributions of the numerator and the denominator of the F-type mixture
respectively, resulting in an approximation F-distribution whose degrees of
freedom can be consistently estimated from the data. The asymptotic power of
the proposed F-type test is established. Two simulation studies are conducted
and they show that in terms of size control, the proposed F-type test
outperforms two existing competitors. The proposed F-type test is also
illustrated by a real data example.
