WIAS Preprint No. 295, (1996)

Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations



Authors

  • Neumann, Michael H.

2010 Mathematics Subject Classification

  • 62G07 62G09 62M07

Keywords

  • Density estimation, strong approximation, bootstrap, weak dependence, mixing, whitening by windowing, simultaneous confidence bands, nonparametric tests

Abstract

We derive a useful approximation of a density estimator based on weakly dependent random vectors by a density estimator built from independent random vectors. We construct, on a sufficiently rich probability space, such a pairing of the random variables of both experiments that the set of observations {X1, ..., Xn} from the time series model is nearly the same as the set of observations {Y1, ... , Yn} from the i.i.d. model. The set ({X1, ... , Xn}Δ{Y1, ... , Yn})∩([a1, b1] x ... x [ad, bd]) has with a high probability at most O({[n1/2 ∏ (bi - ai)] + 1} log(n)) elements. Although this does not imply very much for parametric problems, it has important implications in nonparametric statistics. It yields a strong approximation of a kernel estimator of the stationary density by a kernel density estimator in the i.i.d. model. Moreover, we show that such a strong approximation is also valid for the standard bootstrap and the smoothed bootstrap. Using these results we derive simultaneous confidence bands as well as supremum-type nonparametric tests based on reasoning for the i.i.d. model.

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