ABSTRACT

In this paper, we investigate the problem of estimating the probability density function. The kernel density estimation with bias reduced is nowadays a standard technique in explorative data analysis, there is still a big dispute on how to assess the quality of the estimate and which choice of bandwidth is optimal. This framework examines the most important bandwidth selection methods for kernel density estimation in the context of with bias reduction. Normal reference, least squares cross-validation, biased cross-validation and beta-divergence loss methods are described and expressions are presented. In order to assess the performance of our various bandwidth selectors, numerical simulations and environmental data are carried out.

• Basu, A., Harris, I. R., Hjort, N. L., & Jones, M. C. (1998). Robust and efficient estimation by minimising a density power divergence. Biometrika, 85(3), 549–559. https://doi.org/10.1093/biomet/85.3.549 [Crossref], [Web of Science ®], [Google Scholar]
• Bowman, A. W. (1984). An alternative method of cross-validation for the smoothing of density estimates. Biometrika, 71(2), 353–360. https://doi.org/10.1093/biomet/71.2.353 [Crossref], [Web of Science ®], [Google Scholar]
• Cao, R., Cuevas, A., & Gonalez-Manteiga, W. (1994). A comparative study of several smoothing methods in density estimation? Computational Statistics and Data Analysis, 17(2), 153–176. https://doi.org/10.1016/0167-9473(92)00066-Z [Crossref], [Web of Science ®], [Google Scholar]
• Cichocki, A., Zdunek, R., & Amari, S. (2006). Csiszar's divergences for nonnegative matrix factorization: Family of new algorithms. In Lecture notes in computer science (pp. 32–39). Springer. [Google Scholar]
• Dhaker, H., Ngom, P., Deme, E., & Mbodj, M. (2018). New approach for bandwidth selection in the kernel density estimation based on β-divergence. Journal of Mathematical Sciences: Advances and Applications, 51(1), 57–83. https://doi.org/10.18642/jmsaa_7100121962 [Crossref], [Google Scholar]
• Eguchi, S., & Kano, Y. (2001). Robustifying maximum likelihood estimation (Technical Report). Institute of Statistical Mathematics, June. [Google Scholar]
• Eugene, F. S. (1969). Estimation of a probability density function and its derivatives. The Annals of Mathematical Statistics, 40(4), 1187–1195. https://doi.org/10.1214/aoms/1177697495 [Crossref], [Google Scholar]
• Härdle, W. K. (1991). Smoothing techniques: With implementation in S. Springer Science and Business Media. [Crossref], [Google Scholar]
• 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. https://doi.org/10.1080/01621459.1996.10476701 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
• Jorgensen, B. (1997). The Theory of Dispersion Models. Chapman Hall/CRC Monographs on Statistics and Applied Probability. [Google Scholar]
• Kanazawa, Y. (1993). Hellinger distance and Kullback-Leibler loss for the kernel density estimator. Statistics and Probability Letters, 18(4), 315–321. https://doi.org/10.1016/0167-7152(93)90022-B [Crossref], [Web of Science ®], [Google Scholar]
• Mammen, E., Martinez-Miranda, M. D., Nielsen, J. P., & Sperlich, S. (2011). Do-validation for kernel density estimatio? Journal of the American Statistical Association, 106(494), 651–660. https://doi.org/10.1198/jasa.2011.tm08687 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
• Mammen, E., Martinez-Miranda, M. D., Nielsen, J. P., & Sperlich, S. (2014). Further theoretical and practical insight to the do-validated bandwidth selector. Journal of the Korean Statistical Society, 43(3), 355–365. https://doi.org/10.1016/j.jkss.2013.11.001 [Crossref], [Web of Science ®], [Google Scholar]
• Millimet, D. L., List, J. A., & Stengos, T. (2003). The Environmental Kuznets Curve: Real Progress or Misspecified Models. Review of Economics and Statistics, 85(4), 1038–1047. https://doi.org/10.1162/003465303772815916 [Crossref], [Web of Science ®], [Google Scholar]
• Millimet, D. L., & Stengos, T. (2000). A semiparametric approach to modelling the environmental kuznets curve across U.S. States Department of Economics working paper, Southern Methodist University. [Google Scholar]
• Minami, M., & Eguchi, S. (2002). Robust blind source separation by Beta-divergence. Neural Comput., 14(8), 1859–1886. https://doi.org/10.1162/089976602760128045 [Crossref], [Web of Science ®], [Google Scholar]
• Mugdadi, A. R., & Ibrahim, A. A. (2004). A bandwidth selection for kernel density estimation of functions of random variables. Computational Statistics and Data Analysis, 47(1), 49–62. https://doi.org/10.1016/j.csda.2003.10.013 [Crossref], [Web of Science ®], [Google Scholar]
• Parzen, E. (1962). On estimation of a probability density function and mode. Annals of Mathematical Statistics, 33(3), 1065–1076. https://doi.org/10.1214/aoms/1177704472 [Crossref], [Google Scholar]
• Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. Scandinavia Journal of Statistics, 9(2), 65–78. [Web of Science ®], [Google Scholar]
• Schmalensee, R., Stoker, T. M., & Judson, R. A. (1998). World Carbon Dioxide Emissions, 1950–2050. The Review of Economics and Statistics, 80(1), 15–27. https://doi.org/10.1162/003465398557294 [Crossref], [Web of Science ®], [Google Scholar]
• Scott, W. D. (1992). Multivariate density estimation theory, practice, and visualization. Wiley. [Crossref], [Google Scholar]
• Silverman, B. W. (1986). Density estimation for statistics and data analysis. Chapman and Hall. [Crossref], [Google Scholar]
• Sultan, K. S., & Al-Moisheer, A. S. (2015). Mixture of inverse Weibull and lognormal distributions: Properties, estimation, and illustration. Mathematical Problems in Engineering, 2015. https://doi.org/10.1155/2015/526786 [Crossref], [Web of Science ®], [Google Scholar]
• Taskin, F., & Zaim, O. (2000). Searching for a Kuznets Curve in Environmental Efficiency Using Kernel Estimation. Economics Letters, 68(2), 217–223. https://doi.org/10.1016/S0165-1765(00)00250-0 [Crossref], [Web of Science ®], [Google Scholar]
• Turlach, B. A. (1993). Bandwidth selection in kernel density estimation: A review (Technical Report). Universite catholique de Louvain. [Google Scholar]
• Von Alven, W. H. (Ed.). (1964). Reliability engineering. Prentice Hall. [Google Scholar]
• Wand, M. P., & Jones, M. C. (1995). Kernel smoothing. Chapman and Hall. [Crossref], [Google Scholar]
• Xie, X., & Wu, J. (2014). Some Improvement on Convergence Rates of Kernel Density Estimator. Applied Mathematics, 5(11), 1684–1696. https://doi.org/10.4236/am.2014.511161 [Crossref], [Google Scholar]

To cite this article: Hamza Dhaker, El Hadji Deme & Youssou Ciss (2021) β-divergence loss for the kernel density estimation with bias reduced, Statistical Theory and Related Fields, 5:3, 221-231, DOI: 10.1080/24754269.2020.1858630

To link to this article: https://doi.org/10.1080/24754269.2020.1858630