Bootstrap consistency for general semiparametric M-estimation

Type
Article

Authors
Cheng, Guang
Huang, Jianhua Z.

KAUST Grant Number
KUS-CI-016-04

Date
2010-10

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 onclusions 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 ootstrap weights. This paper provides a first general theoretical study of the bootstrap in semiparametric models. © Institute of Mathematical Statistics, 2010.

Citation
Cheng G, Huang JZ (2010) Bootstrap consistency for general semiparametric M-estimation. The Annals of Statistics 38: 2884–2915. Available: http://dx.doi.org/10.1214/10-AOS809.

Acknowledgements
Supported by NSF Grant DMS-09-06497.Supported in part by NSF Grants DMS-06-06580, DMS-09-07170, NCI Grant CA57030 and Award Number KUS-CI-016-04, made by King Abdullah University of Science and Technology (KAUST).

Publisher
Institute of Mathematical Statistics

Journal
The Annals of Statistics

DOI
10.1214/10-AOS809

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