Semiparametric M-estimation with non-smooth criterion functions

Delsol, Laurent;Van Keilegom, Ingrid
(2020) Annals of the Institute of Statistical Mathematics — Vol. 72, p. 577-605 (2020)

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Abstract
Weareinterestedintheestimationofaparameter θ thatmaximizesacertaincriterion 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 functionsatisfiescertainsmoothnessproperties.Incertainapplications,thesesmoothnessconditionsare,however,notsatisfied.Theaimofthispaperisthereforetoextend the existing theory on semiparametric M-estimators, in order to cover non-smooth M-estimators 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 theseconditionsforaspecificexampleofasemiparametric M-estimatorcomingfrom the area of classification with missing data.
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Delsol, L., & Van Keilegom, I. (2020). Semiparametric M-estimation with non-smooth criterion functions. Annals of the Institute of Statistical Mathematics, 72, 577-605. https://doi.org/10.1007/s10463-018-0700-y (Original work published 2020)