General M-Estimator Processes and their m out of n Bootstrap with Functional Nuisance Parameters

Salim Bouzebda (), Issam Elhattab () and Anouar Abdeldjaoued Ferfache ()
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Salim Bouzebda: Université de technologie de Compiègne
Issam Elhattab: Université de technologie de Compiègne
Anouar Abdeldjaoued Ferfache: Université de technologie de Compiègne

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 4, 2961-3005

Abstract: Abstract In the present paper, we consider the problem of the estimation of a parameter $$\varvec{\theta }$$ θ , in Banach spaces, maximizing some criterion function which depends on an unknown nuisance parameter h, possibly infinite-dimensional. The classical estimation methods are mainly based on maximizing the corresponding empirical criterion by substituting the nuisance parameter by a 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 of this paper 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 the usefulness of our results. Finally, we investigate the performance of the methodology for small samples through a short simulation study.

Keywords: Gaussian process; M-estimation; Empirical process; m out n of bootstrap; Asymptotic distribution; Nuisance parameter; Semiparametric estimation; Non standard distribution; Missing data; Primary: 62G05; 60F17; Secondary: 60F05; 62G09; 62G20; 62H10; 60F15 (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s11009-022-09965-y

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