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Zbl 0970.62038
Groeneboom, Piet; Truax, Donald R.
A monotonicity property of the power function of multivariate tests.
(English)
[J] Indag. Math., New Ser. 11, No.2, 209-218 (2000). ISSN 0019-3577

Summary: Let $S=\sum^n_{k=1} X_kX_k'$, where the $X_k$ are independent observations from a 2-dimensional normal $N(\mu_k, \Sigma)$ distribution, and let $\Lambda= \sum^n_{k=1} \mu_k\mu_k' \Sigma^{-1}$ be a diagonal matrix of the form $\lambda I$, where $\lambda\ge 0$ and $I$ is the identity matrix. It is shown that the density $\varphi$ of the vector $\widetilde \ell=(\ell_1, \ell_2)$ of characteristic roots of $S$ can be written as $G(\lambda,\ell_1, \ell_2)\varphi_0 (\widetilde\ell)$, where $G$ satisfies the FKG condition on $\bbfR^3_+$. This implies that the power function of tests with monotone acceptance region in $\ell_1$ and $\ell_2$, i.e. a region of the form $\{g(\ell_1, \ell_2)\le c\}$, where $g$ is nondecreasing in each argument, is nondecreasing in $\lambda$. It is also shown that the density $\varphi$ of $(\ell_1,\ell_2)$ does not allow a decomposition $\varphi(\ell_1, \ell_2)= G(\lambda,\ell_1, \ell_2)\varphi_0 (\widetilde\ell)$, with $G$ satisfying the FKG condition, if $\Lambda=\text {diag}(\lambda,0)$ and $\lambda> 0$, implying that this approach to proving monotonicity of the power function fails in general.
MSC 2000:
*62H15 Multivariate hypothesis testing
62H10 Multivariate distributions of statistics

Keywords: characteristic roots; FKG condition; power function of tests

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