International E-publication: Publish Projects, Dissertation, Theses, Books, Souvenir, Conference Proceeding with ISBN.  International E-Bulletin: Information/News regarding: Academics and Research

Choice of the bandwidth in Kernel density estimation

Author Affiliations

  • 1Department of Statistics, University of Calcutta, India

Res. J. Mathematical & Statistical Sci., Volume 8, Issue (3), Pages 14-18, September,12 (2020)


Given a set of observations, the knowledge of the underlying probability density function that generates the sample is often of interest. Kernel Density Estimation is a nonparametric method used to guess the underlying density function using the sample observations. Although arguably the most popular method of density estimation, KDE is not free from drawbacks. This method of estimation varies greatly with the choice of the smoothing parameter used to estimate the density. This paper gives an overview of the KDE and discusses some statistical properties of the ideal estimator used to guess the unknown density. An outline of some existing methods of choosing a smoothing parameter are discussed. Here we only consider estimation under the univariate setup. The idea of KDE can easily be generalized to a multivariate dataset.


  1. Silverman, B. W. (1986)., Density estimation for statistics and data analysis., CRC press, pp 1-72. ISBN:0-412-24620-1
  2. Turlach, B. A. (1993)., Bandwidth selection in kernel density estimation: A review., In CORE and Institut de Statistique, 38(3), 1-33.
  3. Devroye, L., & Gyorfi, L. (1985)., Nonparametric density estimation: The L1 view., John Wiley, New York, pp. 191-232. ISBN: 0471-81646-9
  4. Sheather, S. J. (2004)., Density estimation., Statistical science, 19(4), 588-597.
  5. Marron, J. S. and Wand, M. P. (1992)., Exact Mean Integrated Squared Error., Ann. Statist., 20(2), 712-736.
  6. Loader, C. (1999)., Bandwidth Selection: Classical or Plug-In?., The Annals of Statistics, 27(2), 415-438.
  7. Wand, M. P., & Jones, M. C. (1994)., Kernel smoothing., Crc Press, pp 1-85. ISBN:0-412-55270-1,
  8. Rudemo, M. (1982)., Empirical Choice of Histograms and Kernel Density Estimators., Scandinavian Journal of Statistics, 9(2), 65-78.
  9. Jones, M. C., & Kappenman, R. F. (1992)., On a class of kernel density estimate bandwidth selectors., Scandinavian Journal of Statistics,19(4), 337-349.
  10. Jones, M. C., Marron, J. S., & Sheather, S. J. (1996)., A brief survey of bandwidth selection for density estimation., Journal of the American statistical association, 91(433), 401-407.
  11. Stone, C. (1984)., An Asymptotically Optimal Window Selection Rule for Kernel Density Estimates., The Annals of Statistics, 12(4), 1285-1297.
  12. Jones M. C. (1991)., The roles of ISE and MISE in density estimation., Statistics and Probability Letters, 12(1), 51-56.
  13. Devroye, L., & Lugosi, G. (1996)., A universally acceptable smoothing factor for kernel density estimates., The Annals of Statistics, 24(6), 2499-2512.
  14. Yen-Chi Chen (2017)., A tutorial on kernel density estimation and recent advances., Biostatistics & Epidemiology, 1(1), 161-187.
  15. Chu, C. Y., Henderson, D. J., & Parmeter, C. F. (2015)., Plug-in bandwidth selection for kernel density estimation with discrete data., Econometrics, 3(2), 199-214.
  16. Kamalov, F. (2020)., Kernel density estimation based sampling for imbalanced class distribution., Information Sciences, 512, 1192-1201.
  17. Duong, T. (2020)., ks: Kernel density estimation for bivariate data, University of Western Australia, Australia., undefined
  18. Devroye, Luc. (1987)., A Course in Density Estimation. Progress in Probability and Statistics., Birkhauser, Boston, pp 1-35. ISBN:978-0-8176-3365-3.