Skip to main content
Log in

A Bayesian Approach to Bandwidth Selection in Univariate Associate Kernel Estimation

  • Published:
Journal of Statistical Theory and Practice Aims and scope Submit manuscript

Abstract

The fundamental problem in the associate kernel estimation of density or probability mass function (pmf) is the choice of the bandwidth. In this paper, we use a Bayesian approach based upon likelihood cross-validation and a Monte Carlo Markov chain (MCMC) method for deriving the global optimal bandwidth. A comparative simulation study of the MCMC method and the classical methods that adopt the asymptotic mean integrated square error (AMISE) as criterion and the cross validation is presented for data generated from known densities and pmf, using standard AMISE and the practical integrated squared error. The simulation results show the superiority of the MCMC method over the classical methods.

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

Access this article

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

  • Bauwens, L., and M. Lubrano. 1998. Bayesian inference on GARCH models using the Gibbs sampler. Econometrics J., 1, C23–C46.

    Article  Google Scholar 

  • Bowman, A. W. 1984. An alternative method of cross-validation for the smoothing of density estimates. Biometrika, 71, 353–360.

    Article  MathSciNet  Google Scholar 

  • Brewer, M. J. 2000. A Bayesian model for local smoothing in kernel density estimation. Statistics and Computing, 10, 299–309.

    Article  Google Scholar 

  • Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin. 1995. Bayesian data analysis. London, UK, Chapman and Hall.

    MATH  Google Scholar 

  • Gelman, A., G. O. Roberts, and W. R. Gilks. 1996. Efficient metropolis jumping rules. Bayesian Statistics, 5, 599–608.

    MathSciNet  Google Scholar 

  • Gelman, A., and D. B. Rubin. 1992. Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457–511.

    Article  Google Scholar 

  • Hall, P., J. S. Marron, and B. U. Park. 1992. Smoothed cross validation. Probability Theory and Related Fields, 92, 1–20.

    Article  MathSciNet  Google Scholar 

  • Kokonendji, C. C., and T. Senga Kiessé. 2011. Discrete associated kernels method and extensions. Statistical Methodology, 8, 497–516.

    Article  MathSciNet  Google Scholar 

  • Kokonendji, C. C., T. Senga Kiessé, and N. Balakrishnan. 2009. Semiparametric estimation for count data through weighted distributions. J. Stat. Plan. Inf., 139, 3625–3638.

    Article  MathSciNet  Google Scholar 

  • Kokonendji, C. C., T. Senga Kiessé, and S. S. Zocchi. 2007. Discrete triangular distributions and non-parametric estimation for probability mass function. J. Nonparametric Stat., 19, 241–257.

    Article  MathSciNet  Google Scholar 

  • Kokonendji, C. C., and S. S. Zocchi. 2010. Extensions of discrete triangular distributions and boundary bias in kernel estimation for discrete functions. Stat. Prob. Lett., 80, 1655–1662.

    Article  MathSciNet  Google Scholar 

  • Leonard, T. (1978). Density estimation, stochastic processes and prior information. J. R. Stat. Soc. Ser. B, 40, 113–146.

    MathSciNet  MATH  Google Scholar 

  • Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Stat., 33, 1065–1076.

    Article  MathSciNet  Google Scholar 

  • Raftery, A. E., and S. Lewis. 1992. How many iterations in the Gibbs sampler? Bayesian Stat., 4, 763–773.

    Google Scholar 

  • Rosenblatt, M. 1956. Remarks in some nonparametric estimates of a density function. Ann. Math. Stat., 27, 832–837.

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Sheather, S. J., and M. C. Jones. 1991. A reliable data-based bandwidth selection method for kernel density estimation. J. R. Stat. Soc. Ser. B, 53, 683–690.

    MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  • Simonoff, J. S., and G. Tutz. 2000. Smoothing methods for discrete data. In Smoothing and regression: Approaches, computation, and application, ed. M. G. Schimek, 193–228. New York, NY, Wiley.

    MATH  Google Scholar 

  • Terrell, G. R., and D. W. Scott. 1982. Biased and unbiased cross-validation in density estimation. Biometrika, 69, 383–390.

    Article  MathSciNet  Google Scholar 

  • Zhang, X., M. L. King, and R. J. Hyndman. 2006. A bayesian approach to bandwidth selection for multivariate kernel density estimation. Comput. Stat. Data Anal., 50, 3009–3031.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Smail Adjabi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zougab, N., Adjabi, S. & Kokonendji, C.C. A Bayesian Approach to Bandwidth Selection in Univariate Associate Kernel Estimation. J Stat Theory Pract 7, 8–23 (2013). https://doi.org/10.1080/15598608.2013.756286

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1080/15598608.2013.756286

AMS Subject Classification

Keywords

Navigation