Rate of Convergence for the Wild Bootstrap in Nonparametric Regression

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.