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October 2010 Bootstrap consistency for general semiparametric M-estimation
Guang Cheng, Jianhua Z. Huang
Ann. Statist. 38(5): 2884-2915 (October 2010). DOI: 10.1214/10-AOS809

Abstract

Consider M-estimation in a semiparametric model that is characterized by a Euclidean parameter of interest and an infinite-dimensional nuisance parameter. As a general purpose approach to statistical inferences, the bootstrap has found wide applications in semiparametric M-estimation and, because of its simplicity, provides an attractive alternative to the inference approach based on the asymptotic distribution theory. The purpose of this paper is to provide theoretical justifications for the use of bootstrap as a semiparametric inferential tool. We show that, under general conditions, the bootstrap is asymptotically consistent in estimating the distribution of the M-estimate of Euclidean parameter; that is, the bootstrap distribution asymptotically imitates the distribution of the M-estimate. We also show that the bootstrap confidence set has the asymptotically correct coverage probability. These general conclusions hold, in particular, when the nuisance parameter is not estimable at root-n rate, and apply to a broad class of bootstrap methods with exchangeable bootstrap weights. This paper provides a first general theoretical study of the bootstrap in semiparametric models.

Citation

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Guang Cheng. Jianhua Z. Huang. "Bootstrap consistency for general semiparametric M-estimation." Ann. Statist. 38 (5) 2884 - 2915, October 2010. https://doi.org/10.1214/10-AOS809

Information

Published: October 2010
First available in Project Euclid: 20 July 2010

zbMATH: 1200.62042
MathSciNet: MR2722459
Digital Object Identifier: 10.1214/10-AOS809

Subjects:
Primary: 62F40
Secondary: 62G20

Keywords: bootstrap confidence set , Bootstrap consistency , M-estimation , Semiparametric model

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 5 • October 2010
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