Likelihood Ratio Tests for Covariance Structure in Random Effects Models

Abstract

Let W be a p × p matrix distributed according to the Wishart distribution W p ( n, Φ ) with Φ positive definite and n ≥ p. Let (ν n/σ 2) g be distributed according to the chi-squared distribution χ 2(ν n). Consider hierarchical hypotheses H 0 ⊂ H 1 ⊂ H 2 that H 0: Φ = σ 2 I p , H 1: Φ ≥ σ 2 I p , and H 2: Φ , σ 2 are unrestricted. The unbiasedness of the likelihood ratio test (LRT) for testing H 0 against H 1 − H 0 is proved. The LRT for H 1 against H 2 − H 1 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 H 0 against H 1 − H 0 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, Biometrika 52, 447-458).