  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.
