Submitted to the Annals of Statistics BOOTSTRAP CONSISTENCY FOR GENERAL SEMIPARAMETRIC M-ESTIMATION

Consider M-estimation in a semiparametric model that is charac-terized by a Euclidean parameter of interest and an infinite-dimensional nuisance parameter. As a general-purpose approach to statistical in-ferences, the bootstrap has found wide applications in semiparamet-ric M-estimation and, because of its simplicity, provides an attractive alternative to the inference approach based on the asymptotic distri-bution 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 asymp-totically consistent in estimating the distribution of the M-estimate of Euclidean parameter; that is, the bootstrap distribution asymptot-ically imitates the distribution of the M-estimate. We also show that the bootstrap confidence set has the asymptotically correct cover-age 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 bootstra