AISM 52, 139-159

## The local bootstrap for kernel estimators under general dependence conditions

### Efstathios Paparoditis^{1} and Dimitris N. Politis^{2}

^{1}Department of Mathematics and Statistics,
University of Cyprus, P.O. Box 537, CY-1678 Nicosia, Cyprus

^{2}Department 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*
(*X*_{t+1}) | *Y*_{t,m}
= *x*) of a strictly stationary stochastic process
{*X*_{t}, *t* __>__ 1}. In this
notation *phi*(.) is a real-valued Borel
function and *Y*_{t,m}
a segment of lagged values, i.e., *Y*_{t,m} =
(*X*_{t-i1},
*X*_{t-i2},...,
*X*_{t-im}),
where the integers *i*_{j} satisfy
0 __<__ *i*_{1} < *i*_{2}
< ... < *i*_{m} < infinity.
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.

**Source**
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