On the asymptotic behaviour of the pseudolikelihood ratio test statistic with boundary problems

This paper considers the asymptotic distribution of the likelihood ratio statistic T for testing a subset of parameter of interest θ, θ = (γ, η) H₀ γ = γo, based on the pseudolikelihood L(θ,φ) where φ is a consistent estimator of φ, the nuisance parameter. We show that the asymptotic distribution of T under H₀ is a weighted sum of independent chi-squared variables. Some sufficient conditions are provided for the limiting distribution to be a chi-squared variable. When the true value of the parameter of interest θ₀, or the true value of the nuisance parameter φ₀, lies on the boundary of parameter space, the problem is shown to be asymptotically equivalent to the problem of testing the restricted mean of a multivariate normal distribution based on one observation from a multivariate normal distribution with misspecified covariance matrix, or from a mixture of multivariate normal distributions. A variety of examples are provided for which the limiting distributions of T may be mixtures of chi-squared variables. We conducted simulation studies to examine the performance of the likelihood ratio test statistics in variance component models and teratological experiments.