THE LOCAL BOOTSTRAP FOR KERNEL ESTIMATORS UNDER GENERAL DEPENDENCE CONDITIONS*

EFSTATHIOS PAPARODITIS^{1} AND DIMITRIS N. POLITIS^{2}

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; ϕ) = E(ϕ(X_{t+1}) | Y_{t,m} = x) of a strictly stationary stochastic process {X_{t}, t ≥ 1}. In this notation ϕ(·) is a real-valued Borel function and Y_{t,m} a segment of lagged values, i.e., Y_{t,m} = (X_{t−i_{1}}, X_{t−i_{2}}, …, X_{t−i_{m}}), where the integers i_{j} satisfy 0 ≤ i_{1} < i_{2} < ⋯ < i_{m} < ∞. We show that under a fairly weak set of conditions on {X_{t}, t ≥ 1}, an appropriately designed and simple bootstrap procedure that correctly imitates the conditional distribution of X_{t+1} given the selective past Y_{t,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.