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Journal of Statistical Theory and Practice

, Volume 5, Issue 2, pp 303–326 | Cite as

Adaptive Wavelet Estimator for a Function and its Derivatives in an Indirect Convolution Model

  • Christophe Chesneau
Article

Abstract

We consider an indirect convolution model where m blurred and noise-perturbed functions f1,..., fm are randomly observed. For a fixed ω ∈ {f1,..., m}, we want to estimate fω and its derivatives. An adaptive nonlinear wavelet estimator using a singular value decomposition is developed. Taking the mean integrated squared error over Besov balls, we prove that it attains a fast rate of convergence.

AMS Subject Classification

62G07 62G20 

Key-words

Deconvolution Function estimation Rate of convergence Wavelets Hard thresholding 

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References

  1. Bhattacharya, P.K., 1967. Estimation of a probability density function and its derivatives, Sankhya, Serie A, 29, 373–382.MathSciNetzbMATHGoogle Scholar
  2. Cai, T., 1999. Adaptive wavelet estimation: A block thresholding and oracle inequality approach. The Annals of Statistics, 27, 898–924.MathSciNetCrossRefGoogle Scholar
  3. Cai, T., 2002. On adaptive wavelet estimation of a derivative and other related linear inverse problems. Journal of Statistical Planning and Inference, 108, 329–349.MathSciNetCrossRefGoogle Scholar
  4. Cavalier, L., Tsybakov, A.B. 2002. Sharp adaptation for inverse problems with random noise. Probability Theory and Related Fields, 123(3), 323–354.MathSciNetCrossRefGoogle Scholar
  5. Cavalier, L., Raimondo, M. 2007. Wavelet deconvolution with noisy eigenvalues. IEEE Transactions on signal processing, 55, 2414–2424.MathSciNetCrossRefGoogle Scholar
  6. Chaubey, Y.P., Doosti, H., 2005. Wavelet based estimation of the derivatives of a density for m-dependent random variables. Journal of the Iranian Statistical Society, 4(2), 97–105.zbMATHGoogle Scholar
  7. Chaubey, Y.P., Doosti, H., Prakasa Rao, B.L.S., 2006. Wavelet based estimation of the derivatives of a density with associated variables. International Journal of Pure and Applied Mathematics, 27(1), 97–106.MathSciNetzbMATHGoogle Scholar
  8. Chaubey, Y.P., Doosti, H., Prakasa Rao, B.L.S., 2008. Wavelet based estimation of the derivatives of a density for a negatively associated process. Journal of Statistical Theory and Practice, 2(3), 453–463.MathSciNetCrossRefGoogle Scholar
  9. Chesneau, C., 2008. Wavelet estimation via block thresholding: A minimax study under the Lp risk. Statistica Sinica, 183, 1007–1024.MathSciNetzbMATHGoogle Scholar
  10. Chesneau, C., 2010. Wavelet estimation of the derivatives of an unknown function from a convolution model. Current Development in Theory and Applications of Wavelets, 4(2), 131–151.MathSciNetzbMATHGoogle Scholar
  11. Delyon, B., Juditsky, A., 1996. On minimax wavelet estimators. Applied Computational Harmonic Analysis, 3, 215–228.MathSciNetCrossRefGoogle Scholar
  12. Donoho, D. L., Raimondo, M., 2004. Translation invariant deconvolution in a periodic setting. The International Journal of Wavelets, Multiresolution and Information Processing, 2(4), 415–432.MathSciNetCrossRefGoogle Scholar
  13. Fan, J., Koo, J.Y., 2002. Wavelet deconvolution. IEEE transactions on information theory, 48, 734–747.MathSciNetCrossRefGoogle Scholar
  14. Johnstone, I., Kerkyacharian, G., Picard, D., Raimondo, M., 2004. Wavelet deconvolution in a periodic setting. Journal of the Royal Statistical Society, Serie B, 66(3), 547–573.MathSciNetCrossRefGoogle Scholar
  15. Kerkyacharian, G., Picard, D., Raimondo, M., 2007. Adaptive boxcar deconvolution on full Lebesgue measure sets. Statistica Sinica, 17, 317–340.MathSciNetzbMATHGoogle Scholar
  16. Maiboroda, R. E., 1996. Estimators of components of a mixture with varying concentrations. Ukrain. Mat. Zh., 48(4), 562–566.CrossRefGoogle Scholar
  17. Meyer, Y., 1992. Wavelets and Operators. Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  18. Pensky, M., Sapatinas, T., 2009. Functional deconvolution in a periodic setting: uniform case. The Annals of Statistics, 37(1) 73–104.MathSciNetCrossRefGoogle Scholar
  19. Pensky, M., Vidakovic, B., 1999. Adaptive wavelet estimator for nonparametric density deconvolution. The Annals of Statistics, 27, 2033–2053.MathSciNetCrossRefGoogle Scholar
  20. Petrov, V.V., 1995. Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press, Oxford.zbMATHGoogle Scholar
  21. Petsa, A., Sapatinas, T., 2009. Minimax convergence rates under the Lp-risk in the functional deconvolution model. Statistics and Probability Letters, 79, 1568–1576.MathSciNetCrossRefGoogle Scholar
  22. Pokhyl’ko, D., 2005. Wavelet estimators of a density constructed from observations of a mixture. Theory of Probability and Mathematical Statististics, 70, 135–145.CrossRefGoogle Scholar
  23. Prakasa Rao, B.L.S., 1996. Nonparametric estimation of the derivatives of a density by the method of wavelets. Bull. Inform. Cyb., 28, 91–100.MathSciNetzbMATHGoogle Scholar
  24. Prakasa Rao, B.L.S., 2010. Wavelet linear estimation for derivatives of a density from observations of mixtures with varying mixing proportions. Indian Journal of Pure and Applied Mathematics, 41(1), 275–291.MathSciNetCrossRefGoogle Scholar
  25. Rosenthal, H.P., 1970. On the subspaces of Lp (p = 2) spanned by sequences of independent random variables. Israel Journal of Mathematics, 8, 273–303.MathSciNetCrossRefGoogle Scholar
  26. Tsybakov, A., 2004. Introduction à l’estimation nonparamétrique. Springer Verlag, Berlin.Google Scholar
  27. Willer, T., 2005. Deconvolution in white noise with a random blurring effect. LPMA, Preprint.Google Scholar

Copyright information

© Grace Scientific Publishing 2011

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Nicolas OresmeUniversité de Caen Basse-NormandieCaenFrance

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