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

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.


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
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