Minimax and bayes estimation in deconvolution problem*

Mikhail Ermakov

ESAIM: Probability and Statistics (2008)

  • Volume: 12, page 327-344
  • ISSN: 1292-8100

Abstract

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We consider a deconvolution problem of estimating a signal blurred with a random noise. The noise is assumed to be a stationary Gaussian process multiplied by a weight function function εh where h ∈ L2(R1) and ε is a small parameter. The underlying solution is assumed to be infinitely differentiable. For this model we find asymptotically minimax and Bayes estimators. In the case of solutions having finite number of derivatives similar results were obtained in [G.K. Golubev and R.Z. Khasminskii, IMS Lecture Notes Monograph Series36 (2001) 419–433].

How to cite

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Ermakov, Mikhail. "Minimax and bayes estimation in deconvolution problem*." ESAIM: Probability and Statistics 12 (2008): 327-344. <http://eudml.org/doc/250407>.

@article{Ermakov2008,
abstract = { We consider a deconvolution problem of estimating a signal blurred with a random noise. The noise is assumed to be a stationary Gaussian process multiplied by a weight function function εh where h ∈ L2(R1) and ε is a small parameter. The underlying solution is assumed to be infinitely differentiable. For this model we find asymptotically minimax and Bayes estimators. In the case of solutions having finite number of derivatives similar results were obtained in [G.K. Golubev and R.Z. Khasminskii, IMS Lecture Notes Monograph Series36 (2001) 419–433]. },
author = {Ermakov, Mikhail},
journal = {ESAIM: Probability and Statistics},
keywords = {Deconvolution; minimax estimation; Bayes estimation; Wiener filtration; deconvolution},
language = {eng},
month = {5},
pages = {327-344},
publisher = {EDP Sciences},
title = {Minimax and bayes estimation in deconvolution problem*},
url = {http://eudml.org/doc/250407},
volume = {12},
year = {2008},
}

TY - JOUR
AU - Ermakov, Mikhail
TI - Minimax and bayes estimation in deconvolution problem*
JO - ESAIM: Probability and Statistics
DA - 2008/5//
PB - EDP Sciences
VL - 12
SP - 327
EP - 344
AB - We consider a deconvolution problem of estimating a signal blurred with a random noise. The noise is assumed to be a stationary Gaussian process multiplied by a weight function function εh where h ∈ L2(R1) and ε is a small parameter. The underlying solution is assumed to be infinitely differentiable. For this model we find asymptotically minimax and Bayes estimators. In the case of solutions having finite number of derivatives similar results were obtained in [G.K. Golubev and R.Z. Khasminskii, IMS Lecture Notes Monograph Series36 (2001) 419–433].
LA - eng
KW - Deconvolution; minimax estimation; Bayes estimation; Wiener filtration; deconvolution
UR - http://eudml.org/doc/250407
ER -

References

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  1. L.D. Brown, T. Cai, M.G. Low and C. Zang, Asymptotic equivalence theory for nonparametric regression with random design. Ann. Stat.24 (2002) 2399–2430.  
  2. C. Butucea, Deconvolution of supersmooth densities with smooth noise. Canad. J. Statist.32 (2004) 181–192.  Zbl1056.62047
  3. C. Butucea and A.B. Tsybakov, Sharp optimality for density deconvolution with dominating bias. (2004), arXiv:math.ST/0409471.  Zbl1141.62021
  4. L. Cavalier, G.K. Golubev, O.V. Lepski and A.B. Tsybakov, Block thresholding and sharp adaptive estimation in severely ill-posed problems. Theory Probab. Appl.48 (2003) 534–556.  Zbl1130.62313
  5. G.K. Golubev and R.Z. Khasminskii, Statistical approach to Cauchy problem for Laplace equation. State of the Art in Probability and Statistics, Festschrift for W.R. van, Zwet M. de Gunst, C. Klaassen and van der Vaart Eds., IMS Lecture Notes Monograph Series36 (2001) 419–433.  
  6. R.J. Carrol and P. Hall, Optimal rates of convergence for deconvolving a density J. Amer. Statist. Assoc.83 (1988) 1184–1186.  Zbl0673.62033
  7. D.L. Donoho, Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal.2 (1992) 101–126.  Zbl0826.65117
  8. S. Efroimovich, Nonparametric Curve Estimation: Methods, Theory and Applications. New York, Springer (1999).  
  9. S. Efromovich and M. Pinsker, Sharp optimal and adaptive estimation for heteroscedastic nonparametric regression. Statistica Cinica6 (1996) 925–942.  Zbl0857.62037
  10. M.S. Ermakov, Minimax estimation in a deconvolution problem. J. Phys. A: Math. Gen.25 (1992) 1273–1282.  Zbl0765.62080
  11. M.S. Ermakov, Asymptotically minimax and Bayes estimation in a deconvolution problem. Inverse Problems19 (2003) 1339–1359.  Zbl1040.62002
  12. J. Fan, Asymptotic normality for deconvolution kernel estimators. Sankhia Ser. A 53 (1991) 97–110.  Zbl0729.62034
  13. J. Fan, On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist.19 (1991) 1257–1272.  Zbl0729.62033
  14. A. Goldenshluger, On pointwise adaptive nonparametric deconvolution. Bernoulli5 (1999) 907–25.  Zbl0953.62033
  15. Yu K. Golubev, B.Y. Levit and A.B. Tsybakov, Asymptotically efficient estimation of Analitic functions in Gaussian noise. Bernoulli2 (1996) 167–181.  Zbl0860.62034
  16. I.A. Ibragimov and R.Z. Hasminskii, Estimation of distribution density belonging to a class of entire functions. Theory Probab. Appl.27 (1982) 551–562.  Zbl0516.62043
  17. P.A. Jansson, Deconvolution, with application to Spectroscopy. New York, Academic (1984).  
  18. I.M. Johnstone, G. Kerkyacharian, D. Picard and M.Raimondo, Wavelet deconvolution in a periodic setting. J. Roy. Stat. Soc. Ser B.66 (2004) 547–573.  Zbl1046.62039
  19. I.M. Johnstone and M. Raimondo, Periodic boxcar deconvolution and Diophantine approximation. Ann. Statist.32 (2004) 1781–1805.  Zbl1056.62044
  20. J. Kalifa and S. Mallat, Threshholding estimators for linear inverse problems and deconvolutions. Ann. Stat.31 (2003) 58–109.  Zbl1102.62318
  21. S. Kassam and H. Poor, Robust techniques for signal processing. A survey. Proc. IEEE73 (1985) 433–481.  Zbl0569.62084
  22. M.R. Leadbetter, G. Lindgren and H. Rootzen, Extremes and Related Properties of Random sequences and Processes. Springer-Verlag NY (1986).  Zbl0518.60021
  23. R. Neelamani, H. Choi, R.G. Baraniuk, ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems. IEEE Trans. Signal Process.52 (2004) 418–433.  
  24. M. Nussbaum, Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Stat.24 (1996) 2399–2430.  Zbl0867.62035
  25. M. Pensky and B. Vidakovic, Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist.27 (1999) 2033–2053.  Zbl0962.62030
  26. M.S. Pinsker, Optimal filtration of square-integral signal in Gaussian noise. Problems Inform. Transm. 16 (1980) 52–68.  
  27. M. Schipper, Optimal rates and constants in L2-minimax estimation of probability density functions. Math. Methods Stat.5 (1996) 253–274.  Zbl0872.62043
  28. A.J. Smola, B. Scholkopf and K. Miller, The connection between regularization operators and support vector kernels. Newral Networks11 (1998) 637–649.  
  29. A. Tikhonov and V. Arsenin, Solution of Ill-Posed Problems. New-York, Wiley (1977).  Zbl0354.65028
  30. A.B. Tsybakov, On the best rate of adaptive estimation in some inverse problems. C.R. Acad. Sci. Paris, Serie 1330 (2000) 835–840.  Zbl1163.62316
  31. N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series. New York, Wiley (1950).  
  32. * This paper was partially supported by RFFI Grants 02-01-00262, 4422.2006.1.  

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