On the Asymptotic Behaviour of the Pseudolikelihood Ratio Test Statistic

SUMMARY This paper concerns the asymptotic distribution of the likelihood ratio statistic T for testing H 0: θ = θ 0 based on the pseudolikelihood L(θ,ϕ^), where ϕ^ is a simple estimator of ϕ. We show that the asymptotic distribution of T under H 0 is a weighted sum of independent χ 1 2-variables where the weights involve the asymptotic joint covariance matrix of ϕ^ and the score function for θ. Some sufficient conditions are provided for the limiting distribution to be χ 2. The result is extended to allow θ 0 to be a boundary value of the θ parameter space, and ϕ to be misspecified in L(θ, ϕ). We also examine the issue of power loss when ϕ is misspecified in L(θ, ϕ). Several examples including variance component models, multivariate survival models, genetic linkage analysis and the Behrens–Fisher problem are presented to demonstrate the scope of the problems considered and to illustrate the results.