Rate of Convergence for the Wild Bootstrap in Nonparametric Regression

Abstract

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.