Let (\boldsymbol{W}) be a (p \times p) matrix distributed according to the Wishart distribution (W_{p}(n, \boldsymbol{\Phi})) with (\boldsymbol{\Phi}) positive definite and (n \geqslant p). Let (\left(m / \sigma^{2}\right) g) be distributed according to the chi-squared distribution (\chi^{2}(v n)). Consider hierarchical hypotheses (H_{0} \subset) (H_{1} \subset H_{2}) that (H_{0}: \Phi=\sigma^{2} I_{\Gamma}, H_{1}: \Phi \geqslant \sigma^{2} I_{p}), and (H_{2}: \boldsymbol{\Phi}, \sigma^{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).
( 1993 Academic Press. Inc