Unbalanced regression analysis with residuals having a covariance structure of intra-class form

Let Yi be an ni X 1 vector of observations, Xi an ni X p matrix of known values, and beta an unknown p X 1 with the structure Yi = Xi beta + epsilon i, where the covariance matrix of epsilon i is of intra-class form, that is Cov (epsilon i) = sigma2[(1 - rho) Ii + rho e i e i'] where Ii is the ni X ni identity matrix and e i is the ni X 1 vector each element of which is unity. This article develops the maximum likelihood estimators of beta, sigma2, and rho when one observes N pairs (Xi, Yi). This situation arises typically in biological problems where one samples clusters of related organisms. The estimation procedure is illustrated in a commonly occurring genetics situation.