Table of Contents
Advances in Statistics
Volume 2015, Article ID 695904, 11 pages
http://dx.doi.org/10.1155/2015/695904
Research Article

Estimation of the Derivatives of a Function in a Convolution Regression Model with Random Design

1Laboratoire de Mathématiques Nicolas Oresme, Université de Caen, BP 5186, 14032 Caen Cedex, France
2École Supérieure de Commerce IDRAC, 47 rue Sergent Michel Berthet, CP 607, 69258 Lyon Cedex 09, France

Received 8 August 2014; Revised 25 February 2015; Accepted 5 March 2015

Academic Editor: Jos De Brabanter

Copyright © 2015 Christophe Chesneau and Maher Kachour. 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. L. Cavalier and A. Tsybakov, “Sharp adaptation for inverse problems with random noise,” Probability Theory and Related Fields, vol. 123, no. 3, pp. 323–354, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. M. Pensky and T. Sapatinas, “On convergence rates equivalency and sampling strategies in functional deconvolution models,” The Annals of Statistics, vol. 38, no. 3, pp. 1793–1844, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. J.-M. Loubes and C. Marteau, “Adaptive estimation for an inverse regression model with unknown operator,” Statistics & Risk Modeling, vol. 29, no. 3, pp. 215–242, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  4. N. Bissantz and M. Birke, “Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators,” Journal of Multivariate Analysis, vol. 100, no. 10, pp. 2364–2375, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. M. Birke, N. Bissantz, and H. Holzmann, “Confidence bands for inverse regression models,” Inverse Problems, vol. 26, no. 11, Article ID 115020, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. T. Hildebrandt, N. Bissantz, and H. Dette, “Additive inverse regression models with convolution-type operators,” Electronic Journal of Statistics, vol. 8, no. 1, pp. 1–40, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. B. L. S. Prakasa Rao, Nonparametric Functional Estimation, Academic Press, Orlando, Fla, USA, 1983. View at MathSciNet
  8. 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 Publisher · View at Google Scholar
  9. W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation, and Statistical Applications, vol. 129 of Lectures Notes in Statistics, Springer, New York, NY, USA, 1998.
  10. B. Vidakovic, Statistical Modeling by Wavelets, John Wiley & Sons, New York, NY, USA, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  11. T. T. Cai, “On adaptive wavelet estimation of a derivative and other related linear inverse problems,” Journal of Statistical Planning and Inference, vol. 108, no. 1-2, pp. 329–349, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. A. Petsa and T. Sapatinas, “On the estimation of the function and its derivatives in nonparametric regression: a Bayesian testimation approach,” Sankhya A, vol. 73, no. 2, pp. 231–244, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  13. C. Chesneau, “A note on wavelet estimation of the derivatives of a regression function in a random design setting,” International Journal of Mathematics and Mathematical Sciences, vol. 2014, Article ID 195765, 8 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. B. Delyon and A. Juditsky, “On minimax wavelet estimators,” Applied Computational Harmonic Analysis, vol. 3, no. 3, pp. 215–228, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. C. Chesneau, “Wavelet estimation of the derivatives of an unknown function from a convolution model,” Current Development in Theory and Applications of Wavelets, vol. 4, no. 2, pp. 131–151, 2010. View at Google Scholar · View at MathSciNet
  16. M. Pensky and B. Vidakovic, “On non-equally spaced wavelet regression,” Annals of the Institute of Statistical Mathematics, vol. 53, no. 4, pp. 681–690, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. 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
  18. I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992.
  19. S. Mallat, A Wavelet Tour of Signal Processing, Elsevier/Academic Press, Amsterdam, The Netherlands, 3rd edition, 2009. View at MathSciNet
  20. Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, UK, 1992. View at MathSciNet
  21. 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 MathSciNet
  22. A. B. Tsybakov, Introduction à l'Estimation Non Paramétrique, Springer, Berlin, Germany, 2004.
  23. J. Fan and J.-Y. Koo, “Wavelet deconvolution,” IEEE Transactions on Information Theory, vol. 48, no. 3, pp. 734–747, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. M. Pensky and B. Vidakovic, “Adaptive wavelet estimator for nonparametric density deconvolution,” The Annals of Statistics, vol. 27, no. 6, pp. 2033–2053, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  25. Y. P. Chaubey, C. Chesneau, and H. Doosti, “Adaptive wavelet estimation of a density from mixtures under multiplicative censoring,” Statistics: A Journal of Theoretical and Applied Statistics, 2014. View at Publisher · View at Google Scholar
  26. 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 MathSciNet · View at Scopus
  27. D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” Journal of the American Statistical Association, vol. 90, no. 432, pp. 1200–1224, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  28. D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Wavelet shrinkage: asymptopia?” Journal of the Royal Statistical Society Series B: Methodological, vol. 57, no. 2, pp. 301–369, 1995. View at Google Scholar · View at MathSciNet
  29. A. Juditsky and S. Lambert-Lacroix, “On minimax density estimation on R,” Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, vol. 10, no. 2, pp. 187–220, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. A. Delaigle and A. Meister, “Nonparametric function estimation under Fourier-oscillating noise,” Statistica Sinica, vol. 21, no. 3, pp. 1065–1092, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus