On Proole Likelihood

We show that semiparametric proole likelihoods, where the nuisance parameter has been prooled out, behave like ordinary likelihoods in that they have a quadratic expansion. In this expansion the score function and the Fisher information are replaced by the eecient score function and eecient Fisher information, respectively. The expansion may be used, among others, to prove the asymptotic normality of the maximum likelihood estimator, to derive the asymptotic chi-squared distribution of the log likelihood ratio statistic, and to prove the consistency of the observed information as an estimator of the inverse of the asymptotic variance. 1. INTRODUCTION A likelihood function for a low-dimensional parameter can be conveniently visualized by its graph. If the likelihood is smooth, then this is roughly, at least locally, a reversed parabola with its top at the maximum likelihood estimator. The (negative) curvature of the graph at the maximum likelihood estimator is known as the \observed information" and provides an estimate for the inverse of the variance of the maximum likelihood estimator: steep likelihoods yield accurate estimates. The use of the likelihood function in this fashion is not possible for higher-dimensional parameters, and fails particularly for semiparametric models. For example, in semiparamet-ric models the observed information, if it exists, would at best be an innnite-dimensional operator. Frequently, this problem is overcome by using a proole likelihood rather than