  For a multinormal distribution with a $p$-dimensional mean vector
${\mbtheta}$ and an arbitrary unknown dispersion matrix ${\mbSigma}$, Rao ([9],
[10]) proposed two tests for the problem of testing $ H_{0}:{\mbtheta}_{1} =
{\bf 0}, {\mbtheta}_{2} = {\bf 0}, {\mbSigma}~
\hbox{unspecified},~\hbox{versus}~H_{1}:{\mbtheta}_{1} \ne {\bf 0},
{\mbtheta}_{2} ={\bf 0}, {\mbSigma}~\hbox{unspecified}$, where
${\mbtheta}^{'}=({\mbtheta}^{'}_{1},{\mbtheta}^{'}_{2})$. These tests are
referred to as Rao's $W$-test (likelihood ratio test) and Rao's $U$-test
(union-intersection test), respectively. This work is inspired by the
well-known work of Marden and Perlman [6] who claimed that Hotelling's
$T^{2}$-test is admissible while Rao's $U$-test is inadmissible. Both Rao's
$U$-test and Hotelling's $T^{2}$-test can be constructed by applying the
union-intersection principle that incorporates the information
${\mbtheta}_{2}={\bf 0}$ for Rao's $U$-test statistic but does not incorporate
it for Hotelling's $T^{2}$-test statistic. Rao's $U$-test is believed to
exhibit some optimal properties. Rao's $U$-test is shown to be admissible by
fully incorporating the information ${\mbtheta}_{2}={\bf 0}$, but Hotelling's
$T^{2}$-test is inadmissible.
