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**Admissibility of invariant tests in the general multivariate analysis of variance problem.**
*(English)*
Zbl 0598.62006

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 ratio under an invariant generalized prior. Hence the class of tests \(\Phi\) can roughly be defined as being generalized Bayes tests with respect to an invariant generalized prior distribution, altered to reject \(H_ 0\) on the exterior of an arbitrarily chosen set in \({\mathcal C}.\)

The key result is that all tests in \(\Phi\) are admissible among invariant tests, and if \(p\geq m\), \(\Phi\) is minimal complete among invariant tests. In addition, if \(m>1\), it is shown that any acceptance region that is monotone in the largest \(m\wedge p\) eigenvalues of the ”likelihood ratio matrix” is not in \(\Phi\) and hence such tests, including the likelihood ratio test, are inadmissible. Another important result is that tests based on the largest \(m\wedge p\) eigenvalues of \(yS^{-1}y'\) of the form \(\phi (x)=1\) if w(x)\(\not\in C\) and \(\phi (x)=0\) if w(x)\(\in C\), with \(C\in {\mathcal C}\), are admissible among all tests. Such tests include the usual ones based on the largest eigenvalue, the trace, or the determinant of the matrix \(yS^{-1}y'.\)

The case \(q=0\) is the MANOVA case, and the author shows that the class \(\Phi\) suitably redefined is minimal complete among invariant tests (without the restriction that \(p\geq m)\). The author’s numerical calculations indicate that the likelihood ratio test, while inadmissible, performs quite well in practice.

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 ratio under an invariant generalized prior. Hence the class of tests \(\Phi\) can roughly be defined as being generalized Bayes tests with respect to an invariant generalized prior distribution, altered to reject \(H_ 0\) on the exterior of an arbitrarily chosen set in \({\mathcal C}.\)

The key result is that all tests in \(\Phi\) are admissible among invariant tests, and if \(p\geq m\), \(\Phi\) is minimal complete among invariant tests. In addition, if \(m>1\), it is shown that any acceptance region that is monotone in the largest \(m\wedge p\) eigenvalues of the ”likelihood ratio matrix” is not in \(\Phi\) and hence such tests, including the likelihood ratio test, are inadmissible. Another important result is that tests based on the largest \(m\wedge p\) eigenvalues of \(yS^{-1}y'\) of the form \(\phi (x)=1\) if w(x)\(\not\in C\) and \(\phi (x)=0\) if w(x)\(\in C\), with \(C\in {\mathcal C}\), are admissible among all tests. Such tests include the usual ones based on the largest eigenvalue, the trace, or the determinant of the matrix \(yS^{-1}y'.\)

The case \(q=0\) is the MANOVA case, and the author shows that the class \(\Phi\) suitably redefined is minimal complete among invariant tests (without the restriction that \(p\geq m)\). The author’s numerical calculations indicate that the likelihood ratio test, while inadmissible, performs quite well in practice.

### MSC:

62C15 | Admissibility in statistical decision theory |

62J10 | Analysis of variance and covariance (ANOVA) |

62H15 | Hypothesis testing in multivariate analysis |

62C07 | Complete class results in statistical decision theory |

62A01 | Foundations and philosophical topics in statistics |