Annals of Statistics

Rate of Convergence for the Wild Bootstrap in Nonparametric Regression

R. Cao-Abad

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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.

Article information

Ann. Statist., Volume 19, Number 4 (1991), 2226-2231.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G99: None of the above, but in this section

Bootstrap kernel smoothing nonparametric regression


Cao-Abad, R. Rate of Convergence for the Wild Bootstrap in Nonparametric Regression. Ann. Statist. 19 (1991), no. 4, 2226--2231. doi:10.1214/aos/1176348394.

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