AISM 52, 139-159

The local bootstrap for kernel estimators under general dependence conditions

Efstathios Paparoditis1 and Dimitris N. Politis2

1Department of Mathematics and Statistics, University of Cyprus, P.O. Box 537, CY-1678 Nicosia, Cyprus
2Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, Dept 0112, La Jolla, CA 92093-0112, U.S.A.

(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.

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