Application of the Theory of Products of Problems to Certain Patterned Covariance Matrices
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问题乘积理论在具有特定结构的协方差矩阵中的应用
- 研究问题
- 当原假设和备择假设下的协方差矩阵具有特定结构时，如何通过问题分解推导相应的最优统计程序？
- 方法与思路
- 作者证明将多数此类问题转换为无结构协方差问题乘积的定理，并应用于多元方差分析、分类及均值结构检验。
- 材料陈述的贡献
- 摘要称问题乘积理论可用于推导所研究问题的最优程序，并进一步处理均值向量与协方差矩阵同时具有结构的情形。
- 阅读关注点
- 应核对矩阵结构和可交换变量的定义、变换定理的适用范围，以及最优性针对何种损失、检验类别或约束。
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原文摘要
This paper describes a general method for deriving optimal procedures for problems where the covariance matrices are patterned under both null and alternative hypotheses. The pattern considered in this paper was first suggested by Olkin (1970) and is a generalization of the intraclass correlation model of Wilks (1946) and arises in the study of interchangeable random variables. We prove a theorem showing how we can transform most such problems to products of problems where the covariance matrices are unpatterned. This theorem is applied to two problems, the multivariate analysis of variance problem and the multivariate classification problem where in both cases the covariance matrix is assumed patterned. We use theorems about products to derive optimal procedures for these problems. We then look at Olkin's pattern for the mean vector, and show that most problems where both the mean vector and covariance matrix are patterned can be transformed to a product of problems, one of which is trivial. The same two examples are studied where now both mean vectors and covariance matrices are assumed patterned. We also consider the problem of testing that the mean vector is patterned when we know the covariance matrix is.