On pseudolikelihood inference for semiparametric models with boundary problems

Consider a semiparametric model indexed by a Euclidean parameter of interest and an infinite-dimensional nuisance parameter. In many applications, pseudolikelihood provides a convenient way to infer the parameter of interest, where the nuisance parameter is replaced by a consistent estimator. The purpose of this paper is to establish the asymptotic behaviour of the pseudolikelihood ratio statistic under semiparametric models. In particular, we consider testing the hypothesis that the parameter of interest lies on the boundary of its parameter space. Under regularity conditions, we establish the equivalence between the asymptotic distributions of the pseudolikelihood ratio statistic and a likelihood ratio statistic for a normal mean problem with a misspecified covariance matrix. This result holds when the nuisance parameter is estimated at a rate slower than the usual rate in parametric models. We study three examples in which the asymptotic distributions are shown to be mixtures of chi-squared variables. We conduct simulation studies to examine the finite-sample performance of the pseudolikelihood ratio test.