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Neumann, Michael H.

Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations. (English) Zbl 0930.62038

Ann. Stat. 26, No. 5, 2014-2048 (1998).
Summary: We derive an 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 \(\{X_1, \dots, X_n\}\) from the time series model is nearly the same as the set of observations \(\{Y_1, \dots, Y_n\}\) from the i.i.d. model. With a high probability all sets of the form \[ \bigl(\{X_1, \dots, X_n\}\Delta \{Y_1, \dots, Y_n\} \bigr)\cap \bigl( [a_1,b_1] \times \cdots \times [a_d,b_d]\bigr) \] contain no more than \(O(\{[n^{1/2}\prod (b_i-a_i)]+1\}\log(n))\) elements, respectively. 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, it is shown 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.

Cited in 23 Documents

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62G10 Nonparametric hypothesis testing
62G09 Nonparametric statistical resampling methods
60F15 Strong limit theorems

Keywords:

density estimation; weak dependence; mixing; whitening by windowing; time series; strong approximation; bootstrap; simultaneous confidence bands
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