Admissibility of invariant tests in the general multivariate analysis of variance problem
type: Item
instance of: scholarly article
title: Admissibility of invariant tests in the general multivariate analysis of variance problem
zbMATH Open document ID: 0598.62006
author: John I. Marden
published in: The Annals of Statistics
publication date: 1983-01-01T00:00:00Z
review text: This very nice paper gives necessary and sufficient conditions for a test to be admissible among invariant tests in the general multivariate analysis of variance problem. In particular, several popular tests based on the ''likelihood ratio matrix'', including the likelihood ratio test itself, are shown to be inadmissible in certain settings. Specifically, let \(y=(y_ 1,y_ 2)\) be a partitioned \(m\times (p+q)\) multivariate normal matrix with mean matrix \((\mu_ 1,\mu_ 2)\). The rows are independent with common covariance matrix \(\Sigma\). Let S, independent of y, be a \((p+q)\times (p+q)\) Wishart matrix on n degrees of freedom with mean \(\Sigma_ 0\). The problem is to test \(H_ 0: \mu_ 1=0\), \(\mu_ 2=0\) vs. \(H_ A: \mu_ 1\neq 0\), \(\mu_ 2=0\). The problem remains invariant under the group of transformations \(G_ 1\times O(m)\) acting on \(X=(y,S)\) as follows: \[ (A,\Gamma):(y,S)\to (\Gamma yA, A'SA), \] where \(G_ 1\) is the group of \((p+q)\times (p+q)\) nonsingular matrices of the form \[ A=\left( \begin{matrix} A_{11}\\ A_{21}\end{matrix} \begin{matrix} 0\\ A_{22}\end{matrix} \right),\quad A_ 1\quad is\quad p\times p,\quad A_ 2\quad is\quad q\times q, \] and O(m) is the group of \(m\times m\) orthogonal matrices. The maximal invariant parameter is \(\Delta =\lambda^{(m\wedge p)}(\mu_ 1\Sigma^{-1}_{11}\mu_ 1')\), where \(\lambda^{(K)}(M)\) is the vector consisting of the K largest ordered characteristic roots of the symmetric matrix M. Under invariance, the null hypothesis becomes \(H_ 0:\Delta =0.\) A class of invariant tests \(\phi\in \Phi\) is defined as follows: \(\phi (x)=1\) if w(x)\(\in C\), \(\phi (x)=1\) if \(d(x:\pi_ 0,\pi_ 1)>c\), and \(\phi (x)=0\) otherwise. Here \(C\in {\mathcal C}\), the class of closed, convex, and nonincreasing (with respect to weak submajorization) subsets of the range space of \(w(x)=\lambda^{m\wedge (p+q)}(T(I+T)^{-1})\), where \(T=yS^{-1}y'\), and \(d(x:\pi_ 0,\pi_ 1)\) is (almost) equivalent to the posterior odds rati…
Mathematics Subject Classification ID: 62C15
Mathematics Subject Classification ID: 62J10
Mathematics Subject Classification ID: 62H15
Mathematics Subject Classification ID: 62C07
Mathematics Subject Classification ID: 62A01
zbMATH DE Number: 3965162
zbMATH Keywords: orthogonal group
zbMATH Keywords: necessary and sufficient conditions
zbMATH Keywords: invariant tests
zbMATH Keywords: general multivariate analysis of variance
zbMATH Keywords: likelihood ratio test
zbMATH Keywords: multivariate normal
zbMATH Keywords: Wishart matrix
zbMATH Keywords: posterior odds ratio
zbMATH Keywords: invariant generalized prior
zbMATH Keywords: generalized Bayes tests
zbMATH Keywords: minimal complete
zbMATH Keywords: MANOVA
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176346323
OpenAlex ID: W1530539200
DOI: 10.1214/AOS/1176346323
description: scientific article; zbMATH DE number 3965162