含协变量的均值不变检验的容许性
Admissibility of invariant tests for means with covariates
AI总结：
针对多元正态分布中含协变量的均值检验问题，本研究通过充分纳入θ₂=0的信息，证明Rao的U检验（并交检验）是容许的，而Hotelling的T²检验不容许，修正了Marden和Perlman的相关结论。
摘要：
针对具有p维均值向量θ和任意未知色散矩阵Σ的多元正态分布，Rao（[9]，[10]）提出了两种检验方法，用于检验H₀:θ₁=0，θ₂=0，Σ未指定，与H₁:θ₁≠0，θ₂=0，Σ未指定的问题，其中θ'=(θ₁',θ₂')。这些检验分别称为Rao的W检验（似然比检验）和Rao的U检验（并交检验）。本研究受Marden和Perlman[6]的著名工作启发，他们声称Hotelling的T²检验是容许的，而Rao的U检验是不容许的。Rao的U检验和Hotelling的T²检验均可通过并交原理构造，Rao的U检验统计量纳入了θ₂=0的信息，而Hotelling的T²检验统计量未纳入该信息。Rao的U检验被认为具有一些最优性质。通过充分纳入θ₂=0的信息，Rao的U检验被证明是容许的，而Hotelling的T²检验是不容许的。
英文摘要：
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.
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