Rate of convergence for the wild bootstrap in nonparametric regression. (English) Zbl 0745.62038

Cao-Abad, R.

Ann. Stat. 19, No. 4, 2226-2231 (1991).

Summary: This paper concerns the distributions used to construct confidence intervals for the regression function in a nonparametric setup. Some rates of convergence for the normal limit, its plug-in approach and the wild bootstrap are obtained conditionally on the explanatory variable \(X\) and also unconditionally. The bound found for the wild bootstrap approximation is slightly better (by a factor \(n^{-1/45}\)) than the bounds given by the plug-in approach or the CLT for the conditional probability.
On the contrary, the unconditional bounds present a different feature: the rate obtained when approximating by the CLT improves the one given by the plug-in approach by a factor of \(n^{-8/45}\), while this last one performs better than the wild bootstrap approximation and the corresponding ratio is \(n^{-1/45}\). It should be mentioned that these two sequences, especially the last one, tend to zero at an extremely slow rate.

MSC:

62G09	Nonparametric statistical resampling methods
62G07	Density estimation
62G15	Nonparametric tolerance and confidence regions
62E20	Asymptotic distribution theory in statistics
62G20	Asymptotic properties of nonparametric inference

Keywords:

kernel smoothing; nonparametric regression; central limit theorem; confidence intervals; rates of convergence; normal limit; plug-in approach; wild bootstrap approximation; CLT; unconditional bounds