##
**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.

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 |