Critical dimension in profile semiparametric estimation

This paper revisits the classical inference results for profile quasi maximum likelihood estimators (profile MLE) in semiparametric models. We mainly focus on two prominent theorems: the Wilks phenomenon and Fisher expansion for the profile BILE are stated in a new fashion allowing finite samples and model misspecification. The method of study is also essentially different from the usual analysis of the semiparametric problem based on the notion of the hardest parametric submodel. Instead we derive finite sample deviation bounds for the linear approximation error for the gradient of the loglikelihood. This novel approach particularly allows to address the impact of the effective target and nuisance dimension On the accuracy of the results. The obtained nonasymptotic results are surprisingly sharp and yield the classical asymptotic statements including the asymptotic normality and efficiency of the profile MLE. The general results are specified for the important special case of an i.i.d, sample and the analysis is exemplified with a single index model.