Semiparametric M-estimation with non-smooth criterion functions

Abstract

We are interested in the estimation of a parameter θ that maximizes a certain criterion function depending on an unknown, possibly infinite dimensional nuisance parameter h. A common estimation procedure consists in maximizing the corresponding empirical criterion, in which the nuisance parameter is replaced by a nonparametric estimator. In the literature, this research topic, commonly referred to as semiparametric M-estimation, has received a lot of attention in the case where the criterion function M satisfies certain smoothness properties. In certain applications, these smoothness conditions are however not satisfied. The aim of this paper is therefore to extend the existing theory on semiparametric M-estimation problems, in order to cover non-smooth M-estimation problems as well. In particular, we develop `high-level' conditions under which the proposed M-estimator is consistent and has an asymptotic limit. We also check these conditions in detail for a specific example of a semiparametric M-estimation problem, which comes from the area of classification with missing data, and which cannot be dealt with using the existing results in the literature.