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Stochastic Majorization of the Log-Eigenvalues of a Bivariate Wishart Matrix

Michael D. Perlman
Lecture Notes-Monograph Series
Vol. 5, Inequalities in Statistics and Probability (1984), pp. 173-177
Stable URL: http://www.jstor.org/stable/4355497
Page Count: 5
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Stochastic Majorization of the Log-Eigenvalues of a Bivariate Wishart Matrix
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Abstract

Let $l=(l_{1},l_{2})$ and $\lambda =(\lambda _{1},\lambda _{2})$, where $\lambda _{1}\geq \lambda _{2}>0$ are the ordered eigenvalues of S and Σ, respectively, and ${\bf S}\sim W_{2}(n,\Sigma)$ is a bivariate Wishart matrix. Let m = (m1,m2) and $\mu =(\mu _{1},\mu _{2})$, where $m_{i}=\text{log}\ l_{i}$ and $\mu _{i}=\text{log}\ \lambda _{i}$. It is shown that $P_{\mu}\{{\bf m}\not\in B\}$ is Schur-convex in μ whenever B is a Schur-monotone set, i.e. [x ∈ B, x majorizes ${\bf x}^{\ast}$] ⇒ ${\bf x}^{\ast}\in B$. This result implies the unbiasedness and power-monotonicity of a class of invariant tests for bivariate sphericity and other orthogonally invariant hypotheses.

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