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October 2001 Bootstrapping nonparametric density estimators with empirically chosen bandwidths
Peter Hall, Kee-Hoon Kang
Ann. Statist. 29(5): 1443-1468 (October 2001). DOI: 10.1214/aos/1013203461

Abstract

We examine the way in which empirical bandwidth choice affects distributional properties of nonparametric density estimators. Two bandwidth selection methods are considered in detail: local and global plug-in rules. Particular attention is focussed on whether the accuracy of distributional bootstrap approximations is appreciably influenced by using the resample version $\hat{h}*$,rather than the sample version $\hat{h}$, of an empirical bandwidth. It is shown theoretically that,in marked contrast to similar problems in more familiar settings, no general first-order theoretical improvement can be expected when using the resampling version. In the case of local plug-in rules, the inability of the bootstrap to accurately reflect biases of the components used to construct the bandwidth selector means that the bootstrap distribution of $\hat{h}*$ is unable to capture some of the main properties of the distribution of $\hat{h}$. If the second derivative component is slightly undersmoothed then some improvements are possible through using $\hat{h}*$, but they would be difficult to achieve in practice. On the other hand, for global plug-in methods, both $\hat{h}$ and $\hat{h}*$ are such good approximations to an optimal, deterministic bandwidth that the variations of either can be largely ignored, at least at a first-order level.Thus, for quite different reasons in the two cases, the computational burden of varying an empirical bandwidth across resamples is difficult to justify.

Citation

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Peter Hall. Kee-Hoon Kang. "Bootstrapping nonparametric density estimators with empirically chosen bandwidths." Ann. Statist. 29 (5) 1443 - 1468, October 2001. https://doi.org/10.1214/aos/1013203461

Information

Published: October 2001
First available in Project Euclid: 8 February 2002

zbMATH: 1043.62028
MathSciNet: MR1873338
Digital Object Identifier: 10.1214/aos/1013203461

Subjects:
Primary: 62G15
Secondary: 62G20

Keywords: Bootstrap methods , Confidence interval , Edgeworth expansion , kernel methods , nonparametric estimation , plug-in rules , rate of convergence , second-order accuracy , smoothing parameter

Rights: Copyright © 2001 Institute of Mathematical Statistics

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Vol.29 • No. 5 • October 2001
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