Abstract
This paper proposed a nonparametric estimator for probability mass function of multivariate data. The estimator is based on discrete multivariate associated kernel without correlation structure. For the choice of the bandwidth diagonal matrix, we presented the Bayes global method against the likelihood cross-validation one, and we used the Bayesian Markov chain Monte Carlo (MCMC) method for deriving the global optimal bandwidth. We have compared the proposed method with the cross-validation method. The performance of both methods is evaluated under the integrated square error criterion through simulation studies based on for univariate and multivariate models. We also presented applications of the proposed methods to bivariate and trivariate real data. The obtained results show that the Bayes global method performs better than cross-validation one, even for the Poisson kernel which is the very bad discrete
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdous, B., Kokonendji, C. C., & Senga Kiessé, T. (2012). On semiparametric regression for count explanatory variables. Journal of Statistical Planning and Inference, 142, 1537–1548.
Aitchison, J., & Aitken, C. G. G. (1976). Multivariate binary discrimination by the kernel method. Biometrika, 63, 413–420.
Aitchison, J., & Ho, C. H. (1989). The multivariate Poisson-log normal distribution. Biometrika, 76, 643–653.
Bouezmarni, T., & Roumbouts, J. V. K. (2010). Nonparametric density estimation for multivariate bounded data. Journal of Statistical Planning and Inference, 140, 139–152.
Brewer, M. J. (1998). A modelling approach for bandwidth selection in kernel density estimation. In Proceedings of COMPSTAT (pp. 203–208). Heidelberg: Physica Verlag.
Brewer, M. J. (2000). A Bayesian model for local smoothing in kernel density estimation. Statistics and Computing, 10, 299–309.
Chacón, J. E., & Duong, T. (2010). Multivariate plug-in bandwidth selection with unconstrained pilot matrices. Test, 19, 375–398.
Chacón, J. E., & Duong, T. (2011). Unconstrained pilot selectors for smoothed cross-validation. Australian and New Zealand Journal of Statistics, 53, 331–351.
De Lima, M. S., & Atuncar, G. S. (2010). A Bayesian method to estimate the optimal bandwidth for multivariate kernel estimator. Journal of Nonparametric Statistics, 23, 137–148.
Duong, T., & Hazelton, M. L. (2003). Plug-in bandwidth matrices for bivariate kernel density estimation. Journal of Nonparametric Statistics, 15, 17–30.
Duong, T., & Hazelton, M. L. (2005). Cross-validation bandwidth matrices for multivariate kernel density estimation. Scandinavian Journal of Statistics, 32, 485–506.
Gangopadhyay, A. K., & Cheung, K. N. (2002). Bayesian approach to the choice of smoothing parameter in kernel density estimation. Journal of Nonparametric Statistics, 14, 655–664.
Hu, S., Poskitt, D. S., & Zhang, X. (2012). Bayesian adaptive bandwidth kernel density estimation of irregular multivariate distributions. Computational Statistics and Data Analysis, 56, 732–740.
Kokonendji, C. C., & Senga Kiessé, T. (2011). Discrete associated kernels method and extensions. Statistical Methodology, 8, 497–516.
Kokonendji, C. C., Senga Kiessé, T., & Balakrishnan, N. (2009). Semiparametric estimation for count data through weighted distributions. Journal of Statistical Planning and Inference, 139, 3625–3638.
Kokonendji, C. C., Senga Kiessé, T., & Zocchi, S. S. (2007). Discrete triangular distributions and non-parametric estimation for probability mass function. Journal of Nonparametric Statistics, 19, 241–257.
Kokonendji, C. C., & Somé, S. M. (2015). On multivariate associated kernels for smoothing general density function. arXiv: 1502.01173.
Kulasekera, K. B., & Padgett, W. J. (2006). Bayes bandwidth selection in kernel density estimation with censored data. Journal of Nonparametric Statistics, 18, 129–143.
Kuruwita, C. N., Kulasekera, K. B., & Padgett, W. J. (2010). Density estimation using asymmetric kernels and Bayes bandwidths with censored data. Journal of Statistical Planning and Inference, 140, 1765–1774.
Li, Q., & Racine, J. S. (2007). Nonparametric econometrics: Theory and practice. Princeton, Oxford: Princeton University Press.
Racine, J. S., & Li, Q. (2004). Nonparametric estimation of regression functions with both categorical and continuous data. Journal of Econometrics, 119, 99–130.
R Development Core Team (2015). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing, ISBN: 3-900051-07-0, http://www.R-project.org.
Sain, S. R., Baggerly, K. A., & Scott, D. W. (1994). Cross-validation of multivariate densities. Journal of the American Statistical Association, 89, 807–817.
Senga Kiessé, T., Zougab, N., & Kokonendji, C. C. (2015). Bayesian estimation of bandwidth in semiparametric kernel estimation of unknown probability mass and regression functions of count data. Computational Statistics, https://doi.org/10.1007/s00180-015-0627-1.
Silverman, B. W. (1986). Density estimation for statistics and data analysis. London: Chapman and Hall.
Simonoff, J. S. (1996). Smoothing methods in statistics. New York: Springer-Verlag.
Somé, S. M., & Kokonendji, C. C. (2015). Effects of associated kernels in nonparametric multiple regressions. arXiv: 1502.01488.
Wand, M. P., & Jones, M. C. (1994). Multivariate plug-in bandwidth selection. Computational Statistics, 9, 97–116.
Wand, M., & Jones, M. (1995). Kernel smoothing. London: Chapman and Hall.
Wang, M. C., & Ryzin, J. V. (1981). A class of smooth estimators for discrete distributions. Biometrika, 68, 301–309.
Wansouwé, W. E., Kokonendji, C. C., & Kolyang, D. T. (2015). Disake: Discrete associated kernel estimators. URL: http://cran.r-project.org.
Zhang, X., King, M. L., & Hyndman, R. J. (2006). A Bayesian approach to bandwidth selection for multivariate kernel density estimation. Computational Statistics and Data Analysis, 50, 3009–3031.
Zhang, X., King, M. L., & Shang, H. L. (2013). Bayesian bandwidth selection for a nonparametric regression model with mixed types of regressors. https://ideas.repec.org.
Zougab, N., Adjabi, S., & Kokonendji, C. C. (2012). Binomial kernel and Bayes local bandwidth in discrete functions estimation. Journal of Nonparametric Statistics, 24, 783–795.
Zougab, N., Adjabi, S., & Kokonendji, C. C. (2013a). A Bayesian approach to bandwidth selection in univariate associate kernel estimation. Journal of Statistical Theory and Practice, 7, 8–23.
Zougab, N., Adjabi, S., & Kokonendji, C. C. (2013b). Adaptive smoothing in associated kernel discrete functions estiamtion using Bayesian approach. Journal of Statistical Computation and Simulation., 83, 2219–2231.
Zougab, N., Adjabi, S., & Kokonendji, C. C. (2014). Bayesian estimation of adaptive bandwidth matrices in multivariate kernel density estimation. Computational Statistics and Data Analysis, 75, 28–38.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Belaid, N., Adjabi, S., Zougab, N. et al. Bayesian bandwidth selection in discrete multivariate associated kernel estimators for probability mass functions. J. Korean Stat. Soc. 45, 557–567 (2016). https://doi.org/10.1016/j.jkss.2016.04.001
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1016/j.jkss.2016.04.001