## The Annals of Statistics

### Bootstrapping nonparametric density estimators with empirically chosen bandwidths

#### 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.

#### Article information

Source
Ann. Statist. Volume 29, Number 5 (2001), 1443-1468.

Dates
First available in Project Euclid: 8 February 2002

http://projecteuclid.org/euclid.aos/1013203461

Digital Object Identifier
doi:10.1214/aos/1013203461

Mathematical Reviews number (MathSciNet)
MR1873338

Zentralblatt MATH identifier
1043.62028

Subjects
Primary: 62G15: Tolerance and confidence regions
Secondary: 62G20: Asymptotic properties

#### Citation

Hall, Peter; Kang, Kee-Hoon. Bootstrapping nonparametric density estimators with empirically chosen bandwidths. Ann. Statist. 29 (2001), no. 5, 1443--1468. doi:10.1214/aos/1013203461. http://projecteuclid.org/euclid.aos/1013203461.

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