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Journal of Probability and Statistics
Volume 2015 (2015), Article ID 242683, 21 pages
http://dx.doi.org/10.1155/2015/242683
Research Article

Optimal Bandwidth Selection for Kernel Density Functionals Estimation

Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA

Received 10 April 2015; Revised 19 June 2015; Accepted 21 June 2015

Academic Editor: Ricardas Zitikis

Copyright © 2015 Su Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. I. A. Ahmad, “Nonparametric estimation of the location and scale parameters based on density estimation,” Annals of the Institute of Statistical Mathematics, vol. 34, no. 1, pp. 39–53, 1982. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. I. A. Ahmad and M. Amezziane, “Estimation of location and scale parameters based on kernel functional estimators,” in Nonparametric Statistics and Mixture Models: A Festschrift in Honor of Thomas P Hettmansperger, pp. 1–14, World Scientific, 2011. View at Google Scholar
  3. S. Chen, Nonparametric ANOVA using Kernel methods [Ph.D. dissertation], Department of Statistics, Oklahoma State University, 2013.
  4. J. C. Aubuchon and T. Hettmansperger, “A note on the estimation of the integral of f2(x),” Journal of Statistical Planning and Inference, vol. 9, no. 3, pp. 321–331, 1984. View at Publisher · View at Google Scholar
  5. E. L. Lehmann, “Nonparametric confidence intervals for a shift parameter,” The Annals of Mathematical Statistics, vol. 34, pp. 1507–1512, 1963. View at Publisher · View at Google Scholar · View at MathSciNet
  6. R. Grübel, “Estimation of density functionals,” Annals of the Institute of Statistical Mathematics, vol. 46, no. 1, pp. 67–75, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. B. W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman & Hall, London, UK, 1986. View at MathSciNet
  8. A. W. Bowman, “An alternative method of cross-validation for the smoothing of density estimates,” Biometrika, vol. 71, no. 2, pp. 353–360, 1984. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. S. J. Sheather and M. C. Jones, “A reliable data-based bandwidth selection method for kernel density estimation,” Journal of the Royal Statistical Society: Series B, vol. 53, no. 3, pp. 683–690, 1991. View at Google Scholar · View at MathSciNet
  10. A. W. Bowman, “A comparative study of some kernel-based nonparametric density estimators,” Journal of Statistical Computation and Simulation, vol. 21, no. 3-4, pp. 313–327, 1985. View at Publisher · View at Google Scholar
  11. M. C. Jones, J. S. Marron, and S. J. Sheather, “A brief survey of bandwidth selection for density estimation,” Journal of the American Statistical Association, vol. 91, no. 433, pp. 401–407, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. C. R. Loader, “Bandwidth selection: classical or plug-in?” The Annals of Statistics, vol. 27, no. 2, pp. 415–438, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. P. Wand and M. C. Jones, Kernel Smoothing, Chapman and Hall, London, UK, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  14. D. W. Scott and G. R. Terrell, “Biased and unbiased cross-validation in density estimation,” Journal of the American Statistical Association, vol. 82, no. 400, pp. 1131–1146, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. W. Härdle and J. S. Marron, “Optimal bandwidth selection in nonparametric regression function estimation,” The Annals of Statistics, vol. 13, no. 4, pp. 1465–1481, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  16. E. A. Nadaraya, “On the integral mean square error of some nonparametric estimates for the density function,” Theory of Probability and Its Applications, vol. 19, no. 1, pp. 133–141, 1974. View at Publisher · View at Google Scholar
  17. M. Woodroofe, “On choosing a delta-sequence,” The Annals of Mathematical Statistics, vol. 41, pp. 1665–1671, 1970. View at Publisher · View at Google Scholar · View at MathSciNet
  18. P. Hall, J. S. Marron, and B. U. Park, “Smoothed cross-validation,” Probability Theory and Related Fields, vol. 92, no. 1, pp. 1–20, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. H.-G. Müller, “Empirical bandwidth choice for nonparametric kernel regression by means of pilot estimators,” Statistics & Decisions, supplement 2, pp. 193–206, 1985. View at Google Scholar · View at MathSciNet
  20. J. G. Staniswalis, “Local bandwidth selection for kernel estimates,” Journal of the American Statistical Association, vol. 84, no. 405, pp. 284–288, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  21. P. Hall, S. J. Sheather, M. C. Jones, and J. S. Marron, “On optimal data-based bandwidth selection in kernel density estimation,” Biometrika, vol. 78, no. 2, pp. 263–269, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. T. Gasser, A. Kneip, and W. Köhler, “A flexible and fast method for automatic smoothing,” Journal of the American Statistical Association, vol. 86, no. 415, pp. 643–652, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  23. D. Ruppert, S. J. Sheather, and M. P. Wand, “An effective bandwidth selector for local least squares regression,” Journal of the American Statistical Association, vol. 90, no. 432, pp. 1257–1270, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  24. J. Fan and I. Gijbels, Local Polynomial Modeling and Its Application, Chapman & Hall, London, UK, 1996. View at MathSciNet
  25. P. Janssen, J. S. Marron, N. Veraverbeke, and W. Sarle, “Scale measures for bandwidth selection,” Journal of Nonparametric Statistics, vol. 5, no. 4, pp. 359–380, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  26. D. W. Scott, Multivariate Density Estimation: Theory, Practice and Visualization, John Wiley & Sons, New York, NY, USA, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  27. S. J. Sheather, “The performance of six popular bandwidth selection methods on some real datasets,” Computational Statistics, vol. 7, pp. 225–250, 1992. View at Google Scholar
  28. N. Mimoto and R. Zitikis, “Czekanowski's index of overlap, its Lp-type extension, and bias reduction,” American Journal of Mathematical and Management Sciences, vol. 29, no. 1-2, pp. 229–261, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. A. J. Lee, U-Statistics: Theory and Practice, vol. 110, Marcel Dekker, New York, NY, USA, 1990. View at MathSciNet
  30. J. Hayya, D. Armstrong, and N. Gressis, “A note on the ratio of two normally distributed variables,” Management Science, vol. 21, no. 11, pp. 1338–1341, 1975. View at Publisher · View at Google Scholar