Open Access
October 2020 Variational analysis of constrained M-estimators
Johannes O. Royset, Roger J-B Wets
Ann. Statist. 48(5): 2759-2790 (October 2020). DOI: 10.1214/19-AOS1905


We propose a unified framework for establishing existence of nonparametric $M$-estimators, computing the corresponding estimates, and proving their strong consistency when the class of functions is exceptionally rich. In particular, the framework addresses situations where the class of functions is complex involving information and assumptions about shape, pointwise bounds, location of modes, height at modes, location of level-sets, values of moments, size of subgradients, continuity, distance to a “prior” function, multivariate total positivity and any combination of the above. The class might be engineered to perform well in a specific setting even in the presence of little data. The framework views the class of functions as a subset of a particular metric space of upper semicontinuous functions under the Attouch–Wets distance. In addition to allowing a systematic treatment of numerous $M$-estimators, the framework yields consistency of plug-in estimators of modes of densities, maximizers of regression functions, level-sets of classifiers and related quantities, and also enables computation by means of approximating parametric classes. We establish consistency through a one-sided law of large numbers, here extended to sieves, that relaxes assumptions of uniform laws, while ensuring global approximations even under model misspecification.


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Johannes O. Royset. Roger J-B Wets. "Variational analysis of constrained M-estimators." Ann. Statist. 48 (5) 2759 - 2790, October 2020.


Received: 1 May 2018; Revised: 1 September 2019; Published: October 2020
First available in Project Euclid: 19 September 2020

MathSciNet: MR4152120
Digital Object Identifier: 10.1214/19-AOS1905

Primary: 62G07

Keywords: Shape-constrained estimation , variational approximations

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 5 • October 2020
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