Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]