On shift permutation invariance, covariance structures, and Toeplitz matrices

Abstract

The objective of this paper is to present a comprehensive study of shift permutation invariant covariance structures in linear models. It follows that the corresponding covariance matrices belong to the class of symmetric circular Toeplitz matrices. Results for the spectrum of shift permutation invariant covariance matrices of random factors in linear models are given. Among others the results are of use when reparametrization conditions are imposed in order to perform inference based on a unique set of parameters.