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.
"Application of the Theory of Products of Problems to Certain Patterned Covariance Matrices." Ann. Statist. 1 (4) 682 - 699, July, 1973. https://doi.org/10.1214/aos/1176342463