  We discuss a one-sample location test that can be used in the case of
high-dimensional data. For high-dimensional data, the power of Hotelling's test
decrises when the dimension is close to the sample size. To address this loss
of power, some non-exact approaches were proposed, e.g., Dempster (1958, 1960),
Bai and Saranadasa (1996) and Srivastava and Du (2006). In this paper, we focus
on Hotelling's test and Dempster's test. The comparative merits and demerits of
these two tests vary according to the local parameters. In particular, we
consider the situation where it is difficult to determine which test should be
used, that is, where the two tests are asymptotically equivalent in terms of
local power. We propose a new statistic based on the weighted averaging of
Hotelling's $T^2$ statistic and Dempster's statistic that can be applied in
such a situation. Our weight is determined on the basis of the maximum local
asymptotic power on a restricted parameter space that induces local asymptotic
equivalence between Hotelling's test and Dempster's test. In addition, some
good asymptotic properties with respect to the local power are shown. Numerical
results show that our test is more stable than Hotelling's $T^2$ statistic and
Dempster's statistic in most parameter settings.
