Bootstrapping nonparametric density estimators with empirically chosen bandwidths. (English) Zbl 1043.62028

Hall, Peter; Kang, Kee-Hoon

Ann. Stat. 29, No. 5, 1443-1468 (2001).

Summary: 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 \(\widehat{h}^*\), rather than the sample version \(\widehat{h}\), 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 \(\widehat{h}^*\) is unable to capture some of the main properties of the distribution of \(\widehat{h}\).
If the second derivative component is slightly undersmoothed then some improvements are possible through using \(\widehat{h}^*\), but they would be difficult to achieve in practice. On the other hand, for global plug-in methods, both \(\widehat{h}\) and \(\widehat{h}^*\) 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.

MSC:

62G07	Density estimation
62G09	Nonparametric statistical resampling methods
65C60	Computational problems in statistics (MSC2010)
62G15	Nonparametric tolerance and confidence regions
62G20	Asymptotic properties of nonparametric inference

Keywords:

Edgeworth expansions; kernel methods; second-order accuracy; smoothing parameter; rate of convergence

Software:

bootstrap; bootlib; KernSmooth

References:

[1]	Davison, A. C. and Hinkley, D. J. (1997). Bootstrap Methods and Their Applications. Cambridge Univ. Press · Zbl 0886.62001
[2]	Efron, B. and Tibshirani, B. J. (1993). An Introduction to the Bootstrap. Chapman andHall, London. · Zbl 0835.62038
[3]	Faraway, J. J. and Jhun, M. (1990). Bootstrap choice of bandwidth for density estimation. J. Amer. Statist. Assoc. 85 1119-1122. JSTOR: · doi:10.2307/2289609
[4]	Hall, P. (1986). On the number of bootstrap simulations requiredto construct a confidence interval. Ann. Statist. 14 1453-1462. · Zbl 0611.62048 · doi:10.1214/aos/1176350169
[5]	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. · Zbl 0809.62031 · doi:10.1080/02331889108802305
[6]	Hall, P. (1993). On Edgeworth expansion and bootstrap confidence bands in nonparametric curve estimation. J. Roy. Statist. Soc. Ser. B 55 291-304. · Zbl 0780.62040
[7]	Hall, P. and Marron, J. S. (1987). Estimation of integratedsquareddensity derivatives. Statist. Probab. Lett. 6 109-115. [Correction (1988) Statist. Probab. Lett. 7 87.] · Zbl 0628.62029
[8]	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. JSTOR: · Zbl 0699.62020
[9]	Marron, J. S. and Wand, M. P. (1992). Exact mean integratedsquarederror. Ann. Statist. 20 712-736. · Zbl 0746.62040 · doi:10.1214/aos/1176348653
[10]	Scott, D. J. (1992). Multivariate Density Estimation-Theory, Practice, and Visualization. Wiley, New York. · Zbl 0850.62006
[11]	Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman andHall, London. · Zbl 0617.62042
[12]	Taylor, C. C. (1989). Bootstrap choice of the smoothing parameter in kernel density estimation. Biometrika 76 705-712. JSTOR: · Zbl 0678.62042 · doi:10.1093/biomet/76.4.705
[13]	Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman andHall, London. · Zbl 0854.62043