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Kronecker 协方差极大似然估计的存在性与唯一性AI 解读

In matrix-valued datasets the sampled matrices often exhibit correlations among both their rows and their columns. A useful and parsimonious model of such dependence is the matrix normal model, in which the covariances among the elements of a random matrix are parameterized in terms of the Kronecker product of two covariance matrices, one representing row covariances and one representing column covariance. An appealing feature of such a matrix normal model is that the…

多线性形式最大值的尾概率及其应用AI 解读

Let Z be a k­way array consisting of independent standard normal variables. For column vectors h1, …, hk, define a multilinear form of degree k by $(h_1 \otimes \cdots \otimes h_k)' \vec(Z)$. We derive formulas for upper tail probabilities of the maximum of a multilinear form with respect to the hi’s under the condition that the hi’s are unit vectors, and of its standardized statistic obtained by dividing by the norm of Z. We also give formulas for the maximum of a symmetric…

光滑或分片光滑锥备择下卡方混合分布的权重AI 解读

We study the problem of testing a simple null hypothesis about the multivariate normal mean vector against smooth or piecewise smooth cone alternatives. We show that the mixture weights of the $\bar{\chi}^2$ distribution of the likelihood ratio test can be characterized as mixed volumes of the cone and its dual. The weights can be calculated by integration involving the second fundamental form on the boundary of the cone. We illustrate our technique by examples involving a…

两个协方差矩阵相等的单侧检验AI 解读

Let $\mathbf{H}$ and $\mathbf{G}$ be independently distributed according to the Wishart distributions $W_m(M,\Phi)$ and $W_m(N,\Psi)$, respectively. We derive the limiting null distributions of the likelihood ratio criteria for testing $H_0: \Phi = \Psi$ against $H_1 - H_0$ with $H_1: \Phi \geq \Psi$, and for testing $H^{(R)}_0: \Phi \geq \Psi, \operatorname{rank}(\Phi - \Psi) \leq R$ (for given $R$) against $H_1 - H^{(R)}_0$. They are particular cases of the chi-bar-squared…

基于累积卡方统计量最大分量的多重比较程序AI 解读

Multiple comparison procedures based on the maximal component of the cumulative chi-squared statistic are discussed. The Markov properties for the successive components of the cumulative chi-squared statistic are proved and applied for obtaining the p- value of the maximal component in each of the one-way and the two-way analysis of variance models and the two-way contingency table. The power of the multiple comparison procedure is compared with those for some well-known…

基于累积卡方统计量最大分量的多重比较程序AI 解读

Multiple comparison procedures based on the maximal component of the cumulative chi-squared statistic are discussed. The Markov properties for the successive components of the cumulative chi-squared statistic are proved and applied for obtaining the p- value of the maximal component in each of the one-way and the two-way analysis of variance models and the two-way contingency table. The power of the multiple comparison procedure is compared with those for some well-known…