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Chinese Journal of Mathematics
Volume 2016 (2016), Article ID 6204874, 8 pages
http://dx.doi.org/10.1155/2016/6204874
Research Article

A Note on the Adaptive Estimation of a Conditional Continuous-Discrete Multivariate Density by Wavelet Methods

1Laboratoire de Mathématiques Nicolas Oresme, Université de Caen, BP 5186, 14032 Caen Cedex, France
2Mashhad University of Medical Sciences, P.O. Box 91735-951, Mashhad, Iran

Received 11 April 2016; Revised 24 May 2016; Accepted 6 June 2016

Academic Editor: Niansheng Tang

Copyright © 2016 Christophe Chesneau and Hassan Doosti. 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. Q. Li and J. S. Racine, Nonparametric Econometrics: Theory and Practice, Princeton University Press, Princeton, NJ, USA, 2007. View at MathSciNet
  2. N. Akakpo and C. Lacour, “Inhomogeneous and anisotropic conditional density estimation from dependent data,” Electronic Journal of Statistics, vol. 5, pp. 1618–1653, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. G. Chagny, “Warped bases for conditional density estimation,” Mathematical Methods of Statistics, vol. 22, no. 4, pp. 253–282, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. Q. Li and J. Racine, “Nonparametric estimation of distributions with categorical and continuous data,” Journal of Multivariate Analysis, vol. 86, no. 2, pp. 266–292, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. A. Antoniadis, “Wavelets in statistics: a review (with discussion),” Journal of the Italian Statistical Society Series B, vol. 6, pp. 97–144, 1997. View at Google Scholar
  6. W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation, and Statistical Applications, vol. 129 of Lecture Notes in Statistics, Springer, New York, NY, USA, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  7. B. Vidakovic, Statistical Modeling by Wavelets, John Wiley & Sons, New York, NY, USA, 1999.
  8. V. A. Vasiliev, “A truncated estimation method with guaranteed accuracy,” Annals of the Institute of Statistical Mathematics, vol. 66, no. 1, pp. 141–163, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. C. Chesneau, I. Dewan, and H. Doosti, “Nonparametric estimation of a two dimensional continuous-discrete density function by wavelets,” Statistical Methodology, vol. 18, pp. 64–78, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, Pa, USA, 1992.
  11. Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge University Press, 1992. View at MathSciNet
  12. A. Cohen, I. Daubechies, and P. Vial, “Wavelets on the interval and fast wavelet transforms,” Applied and Computational Harmonic Analysis, vol. 1, no. 1, pp. 54–81, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, Elsevier, Amsterdam, The Netherlands, 3rd edition, 2009, with Contributions from Gabriel Peyré.
  14. D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika, vol. 81, no. 3, pp. 425–455, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Density estimation by wavelet thresholding,” The Annals of Statistics, vol. 24, no. 2, pp. 508–539, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. G. Kerkyacharian, D. Picard, L. Birgé et al., “Thresholding algorithms, maxisets and well-concentrated bases,” Test, vol. 9, no. 2, pp. 283–344, 2000. View at Publisher · View at Google Scholar
  17. B. Delyon and A. Juditsky, “On minimax wavelet estimators,” Applied and Computational Harmonic Analysis, vol. 3, no. 3, pp. 215–228, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. P. Ramirez and B. Vidakovic, “Wavelet density estimation for stratified size-biased sample,” Journal of Statistical Planning and Inference, vol. 140, no. 2, pp. 419–432, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. J. Fan, “Local linear regression smoothers and their minimax efficiencies,” The Annals of Statistics, vol. 21, no. 1, pp. 196–216, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Abbaszadeh, C. Chesneau, and H. Doosti, “Multiplicative censoring: estimation of a density and its derivatives under the Lp -risk,” REVSTAT Statistical Journal, vol. 11, no. 3, pp. 255–276, 2013. View at Google Scholar · View at MathSciNet
  21. C. Chesneau, I. Dewan, and H. Doosti, “Nonparametric estimation of a quantile density function by wavelet methods,” Computational Statistics and Data Analysis, vol. 94, pp. 161–174, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. H. P. Rosenthal, “On the subspaces Lpp2 spanned by sequences of independent random variables,” Israel Journal of Mathematics, vol. 8, pp. 273–303, 1970. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. V. V. Petrov, Limit Theorems of Probability Theory: Sequences of Independent Random Variables, vol. 4 of Oxford Studies in Probability, Clarendon Press, Oxford, UK, 1995. View at MathSciNet