Title: General M-estimator processes and their $m$ out of $n$ bootstrap with functional nuisance parameters
Authors: Anouar Abdeldjaoued Ferfache - University de Technologie de Compiegne (France) [presenting]
Salim Bouzebda - Universite de Technologie de Compiegne (France)
Abstract: The focus is on the problem of the estimation of a parameter theta, in Banach spaces, maximizing some criterion function which depends on an unknown nuisance parameter $h$, possibly infinite-dimensional. Classical estimation methods are mainly based on maximizing the corresponding empirical criterion by substituting the nuisance parameter with some nonparametric estimator. We show that the M-estimators converge weakly to maximizers of Gaussian processes under rather general conditions. The conventional bootstrap method fails, in general, to consistently estimate the limit law. We show that the $m$ out of $n$ bootstrap, in this extended setting, is weakly consistent under conditions similar to those required for weak convergence of the M-estimators. The aim is therefore to extend the existing theory on the bootstrap of the M-estimators. Examples of applications from the literature are given to illustrate the generality and usefulness of the results. Finally, we investigate the performance of the methodology for small samples through a short simulation study.