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Bootstrap consistency for general semiparametric \(M\)-estimation. (English) Zbl 1200.62042
Summary: 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 the 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.

MSC:
62G09 Nonparametric statistical resampling methods
62F40 Bootstrap, jackknife and other resampling methods
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
62F10 Point estimation
Software:
BayesDA
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