Optimizing the smoothed bootstrap

  • Suojin Wang
Estimation And Prediction


In this paper we develop the technique of a generalized rescaling in the smoothed bootstrap, extending Silverman and Young's idea of shrinking. Unlike most existing methods of smoothing, with a proper choice of the rescaling parameter the rescaled smoothed bootstrap method produces estimators that have the asymptotic minimum mean (integrated) squared error, asymptotically improving existing bootstrap methods, both smoothed and unsmoothed. In fact, the new method includes existing smoothed bootstrap methods as special cases. This unified approach is investigated in the problems of estimation of global and local functionals and kernel density estimation. The emphasis of this investigation is on theoretical improvements which in some cases offer practical potential.

Key words and phrases

Bootstrap functional estimation kernel density estimation mean integrated squared error mean squared error quantile rescaling smoothing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. De Angelis, D. and Young, G. A. (1992). Smoothing the bootstrap,Internat. Statist. Rev.,60, 45–56.Google Scholar
  2. Efron, B. (1979). Bootstrap methods: Another look at the jackknife,Ann. Statist.,7, 1–26.Google Scholar
  3. Efron, B. (1982).The Jackknife, the Bootstrap and Other Resampling Plans, SIAM, Philadelphia.Google Scholar
  4. Efron, B. and Gong, G. (1983). A leisurely look at the bootstrap, the jackknife and crossvalidation,Amer. Statist.,37, 36–48.Google Scholar
  5. Falk, M. and Reiss, R.-D. (1989). Weak convergence of smoothed and unsmoothed bootstrap quantile estimates,Ann. Probab.,17, 362–371.Google Scholar
  6. Fisher, N. I., Mammen, E. and Marron, J. S. (1994). Testing for multimodality,Comput. Statist. Data Anal.,18, 499–512.Google Scholar
  7. Fryer, M. J. (1976). Some errors associated with the non-parametric estimation of density functions,Journal of the Institute of Mathematics and its Applications,18, 371–380.Google Scholar
  8. Goldstein, L. and Messer, K. (1992). Optimal plug-in estimators for nonparametric functional estimation,Ann. Statist.,20, 1306–1328.Google Scholar
  9. Hall, P. and Martin, M. A. (1988). Exact convergence rate of bootstrap quantile variance estimator,Probab. Theory Related Fields,80, 261–268.Google Scholar
  10. Hall, P., DiCiccio, T. J. and Romano, J. P. (1989). On smoothing and the bootstrap,Ann. Statist.,17, 692–704.Google Scholar
  11. Jones, M. C. (1991). On correcting for variance inflation in kernel density estimation,Comput. Statist. Data Anal.,11, 3–15.Google Scholar
  12. Jones, M. C. and Foster, P. J. (1993). Generalized jackknifing and higher order kernels,Journal of Nonparametric Statistics,3, 81–94.Google Scholar
  13. Jones, M. C. and Sheather, S. J. (1991). Using non-stochastic terms to advantage in estimating integrated squared density derivatives,Statist. Probab. Lett.,11, 511–514.Google Scholar
  14. Silverman, B. W. (1981). Using kernel density estimates to investigate multimodality,J. Roy. Statist. Soc. Ser. B,43, 97–99.Google Scholar
  15. Silverman, B. W. (1986).Density Estimation for Statistics and Data Analysis, Chapman and Hall, London.Google Scholar
  16. Silverman, B. W. and Young, G. A. (1987). The bootstrap: to smooth or not to smooth?Biometrika,74, 469–479.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 1995

Authors and Affiliations

  • Suojin Wang
    • 1
  1. 1.Department of StatisticsTexas A&M UniversityCollege StationU.S.A.

Personalised recommendations