Applications of Products to the Generalized Compound Symmetry Problem
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乘积方法在广义复合对称问题中的应用
- 研究问题
- 当协方差矩阵具有广义复合对称结构时，如何把带结构约束的检验问题转化为无结构约束问题的乘积，以利用已有结果？
- 方法与思路
- 作者扩展已有转换思路，证明将广义复合对称检验转化为无结构问题乘积的定理，并应用于三类多元检验问题。
- 材料陈述的贡献
- 摘要称转换覆盖均值和协方差均受结构约束的问题、仅协方差受约束的 MANOVA，以及另一类由协方差结构产生的问题。
- 阅读关注点
- 核对广义复合对称结构、乘积的数学含义和转换条件，区分本文提供的转换定理与依赖既有工作获得的检验结果。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；转换公式及具体检验结果未提供，不能把结构协方差直接视为面板模型或推断刊本变化。
原文摘要
Arnold (1973) studied testing problems where the covariance matrix is assumed to have the generalized correlation structure under both hypotheses. That paper showed how to transform such problems to "products" of unpatterned problems. This paper extends those results to testing problems where the covariance matrix is assumed to have Geisser's (1963) generalization of the pattern of compound symmetry. We prove theorems indicating how to transform such problems to products of unpatterned problems. These results are then applied to three problems: 1. a general problem where both the mean vectors and covariance matrix are patterned (this problem is general enough to include both the multivariate analysis of variance (MANOVA) and classification problems.); 2. the MANOVA problem when only the covariance matrix is patterned; 3. a problem arising only when the covariance matrix is patterned. In this paper we only show how to transform such problems to products of unpatterned problems that have been studied, since in Arnold (1973) it was showing how to convert results about known problems to results about their product.