Semiparametric M-Estimation with Non-Smooth Criterion Functions

Laurent Delsol (1) , Ingrid van Keilegom (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 UCL - Université Catholique de Louvain = Catholic University of Louvain

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 non-parametric 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 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. Finally, we perform a small simulation study to verify the small sample performance of the proposed estimator, and we briefly describe a number of other situations in which the general theory can be applied, and which are not covered by the existing theory for semiparametric M-estimators.

Mots clés: Semiparametric regression; Nonstandard asymptotics; Nuisance parameter; Missing data; M-estimation; Classification; Empirical processes; Asymptotic distribution

Domaines: Statistiques [math.ST]

Laurent Delsol, Ingrid van Keilegom. Semiparametric M-Estimation with Non-Smooth Criterion Functions. 2015. ⟨hal-01127993⟩