Bootstrap consistency for general semiparametric $M$-estimation

TL;DR

This paper provides the first comprehensive theoretical validation of bootstrap methods for semiparametric $M$-estimation, demonstrating their asymptotic consistency and correct coverage in broad settings.

Contribution

It establishes the asymptotic validity of bootstrap procedures for semiparametric $M$-estimates, including cases with non-root-$n$ estimable nuisance parameters.

Findings

Bootstrap distribution asymptotically imitates the $M$-estimate distribution

Bootstrap confidence sets have correct asymptotic coverage

Results apply to broad classes of bootstrap methods with exchangeable weights

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.