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Lower bounds for bandwidth selection in density estimation
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  • Published: June 1991

Lower bounds for bandwidth selection in density estimation

  • Peter Hall1 &
  • J. S. Marron1 

Probability Theory and Related Fields volume 90, pages 149–173 (1991)Cite this article

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Summary

This paper establishes asymptotic lower bounds which specify, in a variety of contexts, how well (in terms of relative rate of convergence) one may select the bandwidth of a kernel density estimator. These results provide important new insights concerning how the bandwidth selection problem should be considered. In particular it is shown that if the error criterion is Integrated Squared Error (ISE) then, even under very strong assumptions on the underlying density, relative error of the selected bandwidth cannot be reduced below ordern −1/10 (as the sample size grows). This very large error indicates that any technique which aims specifically to minimize ISE will be subject to serious practical difficulties arising from sampling fluctuations. Cross-validation exhibits this very slow convergence rate, and does suffer from unacceptably large sampling variation. On the other hand, if the error criterion is Mean Integrated Squared Error (MISE) then relative error of bandwidth selection can be reduced to ordern −1/2, when enough smoothness is assumed. Therefore bandwidth selection techniques which aim to minimize MISE can be much more stable, and less sensitive to small sampling fluctuations, than those which try to minimize ISE. We feel this indicates that performance in minimizing MISE, rather than ISE, should become the benchmark for measuring performance of bandwidth selection methods.

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References

  • Anderson, G.D.: A comparison of methods for estimating a probability density function. Phd Dissertation, University of Washington, 1969

  • Bickel, P., Ritov, Y.: Estimating integrated squared density derivatives: sharp best order of convergence estimates. Sankhya50-A, 381–393 (1988)

    Google Scholar 

  • Bowman, A.W.: An alternative method of cross-validation for the smoothing of density estimates. Biometrika71, 353–360 (1984)

    Google Scholar 

  • Burkholder, D.L.: Distribution function inequalities for martingales. Ann. Probab.1, 19–42 (1973)

    Google Scholar 

  • Devroye, L., Györfi, L.: Nonparametric density estimation: the L1 View. New York; Wiley 1984

    Google Scholar 

  • Donoho, D., Liu, R.: Geometrizing rates of convergence (unpublished manuscript 1987)

  • Es, B. van.: Likelihood cross-validation bandwidth selection for nonparametric kernel density estimators. J. Nonparamet. Stat. (in press 1991)

  • Fryer, M. J.: A review of some nonparametric methods of density estimation. J. Inst. Math. Appl.20, 335–354 (1977)

    Google Scholar 

  • Hall, P.: Limit theorems for stochastic measures of the accuracy of density estimators. Stochastic Processes Appl.13, 11–25 (1982)

    Google Scholar 

  • Hall, P., Marron, J.S.: Extent to which least-squares cross-validation minimises integrated square error in nonparametric density estimation. Probab. Th. Rel. Fields74, 567–581 (1987a)

    Google Scholar 

  • Hall, P., Marron, J.S.: On the amount of noise inherent in bandwidth selection for a kernel density estimator. Ann. Stat.15, 163–181 (1987b)

    Google Scholar 

  • Hall, P., Marron, J.S.: Estimation of integrated squared density derivatives. Stat. Probab. Lett.6, 109–115 (1987c)

    Google Scholar 

  • Härdle, W., Hall, P., Marron, J.S.: How far are automatically chosen regression smoothers from their optimum?. J. Am. Stat. Assoc.83, 86–95 (1988)

    Google Scholar 

  • Mammen, E.: A short note on optimal bandwidth selection for kernel estimators. Stat. Probab. Lett.9, 23–25 (1988)

    Google Scholar 

  • Marron, J.S.: Convergence properties of an empirical error criterion for multivariate density estimation. J. Multivariate Anal.19, 1–13 (1986)

    Google Scholar 

  • Marron, J.S.: Automatic smoothing parameter selection: A survey. Emp. Econ.13, 187–208 (1988)

    Google Scholar 

  • Marron, J.S.: Comments on a data-driven bandwidth selector. Comp. Stat. Data Anal.8, 155–170 (1989)

    Google Scholar 

  • Marron, J.S., Härdle, W.: Random approximations to some measures of accuracy in nonparametric curve estimation. J. Multivariate Anal.20, 91–113 (1986)

    Google Scholar 

  • Park, B.U., Marron, J.S.: Comparison of data-driven bandwidth selectors. J. Am. Stat. Assoc.85, 66–72 (1990)

    Google Scholar 

  • Rosenblatt, M.: Remarks on some non-parametric estimates of a density function. Ann. Math. Stat.27, 832–837 (1956)

    Google Scholar 

  • Rosenblatt, M.: Curve estimates. Ann. Math. Stat.42, 1815–1842 (1971)

    Google Scholar 

  • Rudemo, M.: Empirical choice of histograms and kernel density estimators. Scand. J. Stat.9, 65–78 (1982)

    Google Scholar 

  • Scott, D.W., Terrell, G.R.: Biased and unbiased cross-validation in density estimation. J. Am. Stat. Assoc.82, 1131–1146 (1987)

    Google Scholar 

  • Silverman, B.W.: Density estimation for statistics and data analysis. New York: Chapman and Hall 1986

    Google Scholar 

  • Steele, J.M.: Invalidity of average squared error criterion in density estimation. Can. J. Stat.6, 193–200 (1978)

    Google Scholar 

  • Stone, C.J.: Optimal convergence rates for nonparametric estimators. Ann. Stat.8, 1348–1360 (1980)

    Google Scholar 

  • Stone, C.J.: Optimal global rates of convergence of nonparametric regression. Ann. Stat.10, 1040–1053 (1982)

    Google Scholar 

  • Watson, G.S., Leadbetter, M.R.: On the estimation of the probability density, I. Ann. Math. Stat.34, 480–491 (1963)

    Google Scholar 

  • Wegman, E.J.: Nonparametric probability density estimation: II. A comparison of density estimation methods. J. Stat. Comput. Simulation1, 225–245 (1972)

    Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, Brown University and University of North Carolina, 27599-3260, Chapel Hill, NC, USA

    Peter Hall & J. S. Marron

Authors
  1. Peter Hall
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  2. J. S. Marron
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Additional information

Research partially supported by National Science Foundation Grants DMS-8701201 and DMS-8902973

Research of the first author was done while on leave from the Australian National University

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Hall, P., Marron, J.S. Lower bounds for bandwidth selection in density estimation. Probab. Th. Rel. Fields 90, 149–173 (1991). https://doi.org/10.1007/BF01192160

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  • Received: 07 June 1989

  • Revised: 07 March 1991

  • Issue Date: June 1991

  • DOI: https://doi.org/10.1007/BF01192160

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Keywords

  • Relative Error
  • Mathematical Biology
  • Selection Problem
  • Kernel Density
  • Strong Assumption
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