Admissibility of invariant tests for means with covariates
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866909408167985152 |
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| author | Tsai, Ming-Tien |
| author_facet | Tsai, Ming-Tien |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1704_00530 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Admissibility of invariant tests for means with covariates Tsai, Ming-Tien Statistics Theory 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. |
| title | Admissibility of invariant tests for means with covariates |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/1704.00530 |