The Annals of Statistics

Bootstrapping nonparametric density estimators with empirically chosen bandwidths

Peter Hall and Kee-Hoon Kang

Full-text: Open access


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

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

First available in Project Euclid: 8 February 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • Davison, A. C. and Hinkley, D. J. (1997). Bootstrap Methods and Their Applications. Cambridge Univ. Press
  • Efron, B. and Tibshirani, B. J. (1993). An Introduction to the Bootstrap. Chapman andHall, London.
  • Faraway, J. J. and Jhun, M. (1990). Bootstrap choice of bandwidth for density estimation. J. Amer. Statist. Assoc. 85 1119-1122.
  • Hall, P. (1986). On the number of bootstrap simulations requiredto construct a confidence interval. Ann. Statist. 14 1453-1462.
  • Hall, P. (1991). Edgeworth expansions for nonparametric density estimators, with applications. Math. Operat. Statistik Ser. Statist. 22 215-232. Hall, P. (1992a). The Bootstrapand Edgeworth Expansion. Springer, New York. Hall, P. (1992b). Effect of bias estimation on coverage accuracy on bootstrap confidence intervals for a probability density. Ann. Statist. 20 675-694.
  • Hall, P. (1993). On Edgeworth expansion and bootstrap confidence bands in nonparametric curve estimation. J. Roy. Statist. Soc. Ser. B 55 291-304.
  • Hall, P. and Marron, J. S. (1987). Estimation of integratedsquareddensity derivatives. Statist. Probab. Lett. 6 109-115. [Correction (1988) Statist. Probab. Lett. 7 87.]
  • Hall, P. and Titterington, D. M. (1989). The effect of simulation order on level accuracy and power of Monte Carlo tests. J. Roy. Statist. Soc. Ser. B 51 459-467.
  • Marron, J. S. and Wand, M. P. (1992). Exact mean integratedsquarederror. Ann. Statist. 20 712-736.
  • Scott, D. J. (1992). Multivariate Density Estimation-Theory, Practice, and Visualization. Wiley, New York.
  • Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman andHall, London.
  • Taylor, C. C. (1989). Bootstrap choice of the smoothing parameter in kernel density estimation. Biometrika 76 705-712.
  • Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman andHall, London.