Admissibility of invariant tests for means with covariates

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.