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A monotonicity property of the power function of multivariate tests. (English) Zbl 0970.62038
Summary: Let $S={\sum }_{k=1}^{n}{X}_{k}{X}_{k}^{\text{'}}$, where the ${X}_{k}$ are independent observations from a 2-dimensional normal $N\left({\mu }_{k},{\Sigma }\right)$ distribution, and let ${\Lambda }={\sum }_{k=1}^{n}{\mu }_{k}{\mu }_{k}^{\text{'}}{{\Sigma }}^{-1}$ 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 $\phi$ of the vector $\stackrel{˜}{\ell }=\left({\ell }_{1},{\ell }_{2}\right)$ of characteristic roots of $S$ can be written as $G\left(\lambda ,{\ell }_{1},{\ell }_{2}\right){\phi }_{0}\left(\stackrel{˜}{\ell }\right)$, where $G$ satisfies the FKG condition on ${ℝ}_{+}^{3}$. 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 $\left\{g\left({\ell }_{1},{\ell }_{2}\right)\le c\right\}$, where $g$ is nondecreasing in each argument, is nondecreasing in $\lambda$. It is also shown that the density $\phi$ of $\left({\ell }_{1},{\ell }_{2}\right)$ does not allow a decomposition $\phi \left({\ell }_{1},{\ell }_{2}\right)=G\left(\lambda ,{\ell }_{1},{\ell }_{2}\right){\phi }_{0}\left(\stackrel{˜}{\ell }\right)$, with $G$ satisfying the FKG condition, if ${\Lambda }=\text{diag}\left(\lambda ,0\right)$ and $\lambda >0$, implying that this approach to proving monotonicity of the power function fails in general.
##### MSC:
 62H15 Multivariate hypothesis testing 62H10 Multivariate distributions of statistics