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., Yt,m=(Xt-i 1,Xt-i 2,...,Xt-i m), where the integers i i , satisfy 0 ≤ i1<i2...<im∞>. 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.
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Paparoditis, E., Politis, D.N. The Local Bootstrap for Kernel Estimators under General Dependence Conditions. Annals of the Institute of Statistical Mathematics 52, 139–159 (2000). https://doi.org/10.1023/A:1004193117918
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DOI: https://doi.org/10.1023/A:1004193117918