Bootstrap Consistency for General Semiparametric M-estimation

Consider M-estimation in a semiparametric model that is characterized by a Euclidean parameter of interest and a nuisance function parameter. We show that, under general conditions, the bootstrap is asymptotically consistent in estimating the distribution of the M-estimate of Euclidean parameter; this 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. Our results provide a theoretical justification for the use of bootstrap as an inference tool in semiparametric modelling and apply to a broad class of bootstrap methods with exchangeable bootstrap weights. A by-product of our theoretical development is the second order asymptotic linear expansion of the (bootstrap) M-estimate.