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, Φ) with Φ positive definite and n ≥ p. Let (νn/σ2) g be distributed according to the chi-squared distribution χ2(νn). Consider hierarchical hypotheses H0 ⊂ H1 ⊂ H2 that H0: Φ = σ2Ip, H1: Φ ≥ σ2Ip, and H2: Φ, σ2 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).