Principal components and model selection

Let the kp-variate random vector X be partitioned into k subvectors X _i of dimension p each, and let the covariance matrix Ψ of X be partitioned analogously into submatrices Ψ _ij . Based on principal component analysis we suggest a hierarchy of models, where the lowest level assumes independence of all X _i , with identical covariance matrices, and the highest level makes no assumptions about Ψ beyond positive definiteness. The intermediate levels are characterized by a common orthogonal matrix ß which diagonalizes Ψ _ij for all pairs (i, j), i.e., Ψ _ij = ßΛ _ij ß′, where Λ _ij is diagonal. The hierarchy is motivated by both a practical example and theoretical arguments. Model selection based on likelihood ratio tests and information criteria is discussed.