Likelihood Ratio Tests for Covariance Structure in Random Effects Models

Let W be a p - p matrix distributed according to the Wishart distribution Wp(n, [Phi]) with [Phi] positive definite and n >= p. Let ([nu]n/[sigma]2) g be distributed according to the chi-squared distribution [chi]2([nu]n). Consider hierarchical hypotheses H0 [subset of] H1 [subset of] H2 that H0: [Phi] = [sigma]2Ip, H1: [Phi] >= [sigma]2Ip, and H2: [Phi], [sigma]2 are unrestricted. The unbiasedness of the likelihood ratio test (LRT) for testing H0 against H1 - H0 is proved. The LRT for H1 against H2 - H1 is shown to have monotonic property of its power function but not unbiased. As n goes to infinity, limiting null distributions of these two LRT statistics are obtained as mixtures of chi-squared distributions. For a general class of tests for H0 against H1 - H0 including LRT, the local unbiasedness is proved using FKG inequality. Here a new sufficient condition for the FKG condition is posed. These LRTs are shown to have applications to the random effects models introduced by C. R. Rao (1965, Biometrika52, 447-458).