The local bootstrap for kernel estimators under general dependence conditions

Abstract

We consider the problem of estimating the distribution of a nonparametric (kernel) estimator of the conditional expectation g(Greek cursive chi

φ) = E(φ(Xt+1) | Yt,m = Greek cursive chi) of a strictly stationary stochastic process {Xt, t ≥ 1}. In this notation φ(·) is a real-valued Borel function and Yt,m a segment of lagged values, i.e., Yt,m = (Xt-i1, Xt-i2, . . . , Xt-im), where the integers ij satisfy 0 ≤ i1 < i2 < ··· < im < ∞. We show that under a fairly weak set of conditions on {Xt, t ≥ 1}, an appropriately designed and simple bootstrap procedure that correctly imitates the conditional distribution of Xt+1 given the selective past Yt,m, approximates correctly the distribution of the class of nonparametric estimators considered. The procedure proposed is entirely nonparametric, its main dependence assumption refers to a strongly mixing process with a polynomial decrease of the mixing rate and it is not based on any particular assumptions on the model structure generating the observations.