Critical dimension in profile semiparametric estimation

TL;DR

This paper provides new finite sample deviation bounds and nonasymptotic results for profile MLE in semiparametric models, addressing model misspecification and effective dimension issues, and confirming classical asymptotic properties.

Contribution

It introduces a novel approach based on deviation bounds for the gradient's linear approximation, allowing finite sample analysis and addressing model complexity.

Findings

Finite sample deviation bounds for profile MLE

Classical asymptotic normality and efficiency are confirmed

Effective dimension impacts estimation accuracy

Abstract

This paper revisits the classical inference results for profile quasi maximum likelihood estimators (profile MLE) in the semiparametric estimation problem. We mainly focus on two prominent theorems: the Wilks phenomenon and Fisher expansion for the profile MLE 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 important issue of the effective target and nuisance dimension. 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 to the important special cases of an i.i.d. sample and the analysis is exemplified with a single index model.