Bootstrap consistency for general semiparametric $M$-estimation

By Guang Cheng, Jianhua Z. Huang. 2009

157 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/fd165a2bdaa4a9209e8736980cf7411cdbf56d37

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Citations in the corpus

A Note on Conditional Versus Joint Unconditional Weak Convergence in Bootstrap Consistency Results (2017, methodology): “…for the general empirical process based on i.i.d. observations using results of van der Vaart and Wellner (2000, Section 2.9) (note that multiplier CLTs are sometimes also referred to as multiplier or weighted bootstraps; see, e.g., Kosorok, 2008, Cheng and Huang (2010) and the references therein).”
Efficient Quantile Regression Analysis With Missing Observations (2015, background): “Cheng and Huang (2010) demonstrated the bootstrap consistency for general semiparametric M-estimation which includes (12) as a special case.”