Skip to main content
SpringerLink
Log in

Accuracy of transformed kernel density estimates for a heavy-tailed distribution

  • Stochastic Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

Nonparametric estimation for the density of a heavy-tailed probability distribution is studied through transformation of initial observations. The accuracy of transformed kernel estimates with constant and variable window width in the sense of mean integrated squared error for different transformations is determined. Boundary kernel are designed for improving estimation on distribution tails. For a kernel estimate with variable window width, the mismatch method ensures a mean integrated squared estimation error close to the optimal error.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Asmussen, S., Klüppelberg, C., and Sigman, K., Sampling in Subexponential Times, with Queueing Applications, Stochastic Proc. Appl., 1999, no. 79, pp. 265–286.

  2. Silverman, B.W., Density Estimation for Statistics and Data Analysis, New York: Chapman, 1986.

    Google Scholar 

  3. Breiman, L., Meisel, W., and Purcell, E., Variable Kernel Estimates of Multivariate Densities, Technometrics, 1977, no. 19, pp. 135–144.

  4. Abramson, I.S., On Bandwidth Estimation in Kernel Estimators: A Square Root Law, Ann. Statist., 1982, no. 10, pp. 1217–1223.

  5. Hall, P., On Global Properties of Variable Bandwidth Density Estimators, Ann. Statist., 1992, no. 20(2), pp. 762–778.

    Google Scholar 

  6. Naito, K., On a Certain Class of Nonparametric Density Estimators with Reduced Bias, Statist. Prob. Lett., 2001, no. 51, pp. 71–78.

  7. Novak, S.Y., Generalised Kernel Density Estimator, Theory Probab. Appl., 1999, no. 44 (3), pp. 570–583.

  8. Devroye, L. and Györfi, L., Nonparametric Density Estimation. The L1View., New York: Wiley, 1985. Translated under the title Neparametricheskoe otsenivanie plotnosti. L1-podkhod, Moscow: Mir, 1988.

    Google Scholar 

  9. Wand, M.P., Marron, J.S., and Ruppert, D., Transformations in Density Estimation, J. Am. Statist. Assoc., Theory, Methods, 1991, no. 86(414), pp. 343–353.

  10. Yang, L. and Marron, J.S., Iterated Transformation-Kernel Density Estimation, J. Am. Statist. Assoc., 1999, no. 94(446), pp. 580–589.

    Google Scholar 

  11. Markovitch, N.M. and Krieger, U.R., Nonparametric Estimation of Long-Tailed Density Functions and Its Application to Analysis of World Wide Web Traffic, Performance Evaluation, 2000, no. 42(2–3), pp. 205–222.

  12. Markovich, N.M., Transformed Estimates of Densities of Heavy-Tailed Distributions and Classiffication, Avtom. Telemekh, 2002, no. 4, pp. 118–132.

  13. Maiboroda, R.E. and Markovich, N.M., Estimation of Heavy-Tailed Probability Density Function with Application to Web Data, Comput. Stat., 2004, no. 19, pp. 569–592.

  14. Hall, P. and Marron, J.S., Variable Window Width Kernel Estimates of Probability Densities, Prob. Theory Related Fields, 1988, no. 80, pp. 37–49.

  15. Scott, D.W., Multivariate Density Estimation Theory, Practice and Visualization, New York: Wiley, 1992.

    Google Scholar 

  16. Chow, Y.-S., Geman, S., and Wu, L.-D., Consistent Cross-Validated Density Estimation, Ann. Statist., 1983, no. 11, pp. 25–38.

  17. Markovich, N.M., Experimental Analysis of Nonparametric Probability Density Estimates and of Methods for Smoothing Them, Autom. Remote Control, 1989, no. 50, pp. 941–948.

  18. Prakasa Rao, B.L.S., Nonparametric Functional Estimation, New York: Academic, 1983.

    Google Scholar 

  19. Mikosch, T., Regular Variation, Subexponentiality and Their Applications in Probability Theory, Report 99-013, University of Groningen, 1999.

  20. Jurečková, J. and Picek, J., A Class of Tests on the Tail Index, Extremes, 2001, no. 4, pp. 165–183.

  21. Schuster, J., Incorporating Support Constraints into Nonparametric Estimators of Densities, Commun. Stat. Theory Meth., 1985, no. 14(5), pp. 1123–1136.

  22. Simonoff, J.S., Smoothing Methods in Statistics, New York: Springer, 1996.

    Google Scholar 

  23. Hill, B.M., A Simple General Approach to Inference about the Tail of a Distribution, Ann. Statist., 1975, no. 3, pp. 1163–1174.

  24. Resnick, S. and Starica, C., Smoothing the Moment Estimate of the Extreme Value Parameter, Extremes, 1999, no. 1(3), pp. 263–294.

  25. Dekkers, A.L., Einmahl, J.H.J., and de Haan, L., A Moment Estimator for the Index of an Extreme-Value Distribution, Ann. Statis., 1989, no. 17(4), pp. 1833–1855.

    Google Scholar 

  26. Aidu, F.A. and Vapnik, V.N., Probability Density Estimation via Statistical Regularization, Avtom. Telemekh., 1989, no. 4, pp. 84–97.

  27. Vapnik, V.N., Markovich, N.M., and Stephanyuk, A.R., Rate of Convergence in L2 of the Projection Estimator of the Distribution Density, Autom. Remote Control, 1992, no. 53, pp. 677–686.

Download references

Author information

Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    N. M. Markovich

Authors
  1. N. M. Markovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Avtomatika i Telemekhanika, No. 2, 2005, pp. 55–72.

Original Russian Text Copyright © 2005 by Markovich.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Markovich, N.M. Accuracy of transformed kernel density estimates for a heavy-tailed distribution. Autom Remote Control 66, 217–232 (2005). https://doi.org/10.1007/s10513-005-0046-9

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10513-005-0046-9

Keywords

Search

Navigation