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.
"Applications of Products to the Generalized Compound Symmetry Problem." Ann. Statist. 4 (1) 227 - 233, January, 1976. https://doi.org/10.1214/aos/1176343357