Table of Contents Author Guidelines Submit a Manuscript
Journal of Probability and Statistics
Volume 2009, Article ID 275308, 19 pages
http://dx.doi.org/10.1155/2009/275308
Research Article

Asymptotically Sufficient Statistics in Nonparametric Regression Experiments with Correlated Noise

Department of Applied Probability and Statistics, University of California, Santa Barbara, Santa Barbara, CA 93106-3110, USA

Received 30 July 2009; Accepted 16 December 2009

Academic Editor: A. T. A. Wood

Copyright © 2009 Andrew V. Carter. 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. D. Brown and M. G. Low, “Asymptotic equivalence of nonparametric regression and white noise,” The Annals of Statistics, vol. 24, no. 6, pp. 2384–2398, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M. Nussbaum, “Asymptotic equivalence of density estimation and Gaussian white noise,” The Annals of Statistics, vol. 24, no. 6, pp. 2399–2430, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. S. Pinsker, “Optimal filtration of square-integrable signals in Gaussian noise,” Problems of Information Transmission, vol. 16, no. 2, pp. 120–133, 1980. View at Google Scholar · View at MathSciNet
  4. L. D. Brown, T. T. Cai, M. G. Low, and C.-H. Zhang, “Asymptotic equivalence theory for nonparametric regression with random design,” The Annals of Statistics, vol. 30, no. 3, pp. 688–707, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. V. Carter, “A continuous Gaussian approximation to a nonparametric regression in two dimensions,” Bernoulli, vol. 12, no. 1, pp. 143–156, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A. V. Carter, “Asymptotic approximation of nonparametric regression experiments with unknown variances,” The Annals of Statistics, vol. 35, no. 4, pp. 1644–1673, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. Rohde, “On the asymptotic equivalence and rate of convergence of nonparametric regression and Gaussian white noise,” Statistics & Decisions, vol. 22, no. 3, pp. 235–243, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. Reiß, “Asymptotic equivalence for nonparametric regression with multivariate and random design,” The Annals of Statistics, vol. 36, no. 4, pp. 1957–1982, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. I. M. Johnstone and B. W. Silverman, “Wavelet threshold estimators for data with correlated noise,” Journal of the Royal Statistical Society. Series B, vol. 59, no. 2, pp. 319–351, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. I. M. Johnstone, “Wavelet shrinkage for correlated data and inverse problems: adaptivity results,” Statistica Sinica, vol. 9, no. 1, pp. 51–83, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. Wang, “Function estimation via wavelet shrinkage for long-memory data,” The Annals of Statistics, vol. 24, no. 2, pp. 466–484, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. D. L. Donoho and I. M. Johnstone, “Asymptotic minimaxity of wavelet estimators with sampled data,” Statistica Sinica, vol. 9, no. 1, pp. 1–32, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. L. Le Cam, “Sufficiency and approximate sufficiency,” Annals of Mathematical Statistics, vol. 35, pp. 1419–1455, 1964. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. L. Le Cam, Asymptotic Methods in Statistical Decision Theory, Springer Series in Statistics, Springer, New York, NY, USA, 1986. View at MathSciNet
  15. L. Le Cam, “On the information contained in additional observations,” The Annals of Statistics, vol. 2, pp. 630–649, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. Kullback, “A lower bound for discrimination in terms of variation,” IEEE Transactions on Information Theory, vol. 13, pp. 126–127, 1967. View at Google Scholar
  17. I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1992. View at MathSciNet
  18. Y. Meyer, Ondelettes et Opérateurs. I, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, France, 1990. View at MathSciNet
  19. J. Zhang and G. Walter, “A wavelet-based KL-like expansion for wide-sense stationary random processes,” IEEE Transactions on Signal Processing, vol. 42, no. 7, pp. 1737–1745, 1994. View at Publisher · View at Google Scholar · View at Scopus
  20. L. Cavalier, “Estimation in a problem of fractional integration,” Inverse Problems, vol. 20, no. 5, pp. 1445–1454, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. G. W. Wornell, “Karhunen-Loève-like expansion for 1/f processes via wavelets,” IEEE Transactions on Information Theory, vol. 36, no. 4, pp. 859–861, 1990. View at Publisher · View at Google Scholar · View at Scopus
  22. E. J. McCoy and A. T. Walden, “Wavelet analysis and synthesis of stationary long-memory processes,” Journal of Computational and Graphical Statistics, vol. 5, no. 1, pp. 26–56, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  23. D. L. Donoho, “Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition,” Applied and Computational Harmonic Analysis, vol. 2, no. 2, pp. 101–126, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. P. Abry and D. Veitch, “Wavelet analysis of long-range-dependent traffic,” IEEE Transactions on Information Theory, vol. 44, no. 1, pp. 2–15, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. D. Veitch and P. Abry, “A wavelet-based joint estimator of the parameters of long-range dependence,” IEEE Transactions on Information Theory, vol. 45, no. 3, pp. 878–897, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. S. Stoev, V. Pipiras, and M. S. Taqqu, “Estimation of the self-similarity parameter in linear fractional stable motion,” Signal Processing, vol. 82, no. 12, pp. 1873–1901, 2002. View at Publisher · View at Google Scholar · View at Scopus
  27. S. G. Mallat, “Multiresolution approximations and wavelet orthonormal bases of L2(),” Transactions of the American Mathematical Society, vol. 315, no. 1, pp. 69–87, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet