AISM 52, 139-159
(Received October 6, 1997; revised September 14, 1998)
Abstract. We consider the problem of estimating the distribution of a nonparametric (kernel) estimator of the conditional expectation g(x ; phi) = E(phi (Xt+1) | Yt,m = x) of a strictly stationary stochastic process {Xt, t > 1}. In this notation phi(.) 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 < infinity. 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.
Key words and phrases: Resampling, confidence intervals, dependence, nonparametric estimators.