On Profile 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.

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