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Minimax and bayes estimation in deconvolution problem*

Published online by Cambridge University Press:  08 May 2008

Mikhail Ermakov
Mikhail Ermakov*
Affiliation:
Mechanical Engineering Problems Institute, Mechanical Engineering Problems Institute, Russian Academy of Sciences, Bolshoy pr. VO 61, 199178 St.Petersburg, Russia.
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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].

Keywords

Deconvolutionminimax estimationBayes estimationWiener filtration
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 327 - 344
Copyright
© EDP Sciences, SMAI, 2008

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References

Brown, L.D., Cai, T., Low, M.G. and Zang, C., Asymptotic equivalence theory for nonparametric regression with random design. Ann. Stat. 24 (2002) 2399–2430.
Butucea, C., Deconvolution of supersmooth densities with smooth noise. Canad. J. Statist. 32 (2004) 181–192. CrossRef
C. Butucea and A.B. Tsybakov, Sharp optimality for density deconvolution with dominating bias. (2004), arXiv:math.ST/0409471.
Cavalier, L., Golubev, G.K., Lepski, O.V. and Tsybakov, A.B., Block thresholding and sharp adaptive estimation in severely ill-posed problems. Theory Probab. Appl. 48 (2003) 534–556.
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 Series 36 (2001) 419–433.
R.J. Carrol and P. Hall, Optimal rates of convergence for deconvolving a density J. Amer. Statist. Assoc. 83 (1988) 1184–1186.
Donoho, D.L., Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 (1992) 101–126. CrossRef
S. Efroimovich, Nonparametric Curve Estimation: Methods, Theory and Applications. New York, Springer (1999).
Efromovich, S. and Pinsker, M., Sharp optimal and adaptive estimation for heteroscedastic nonparametric regression. Statistica Cinica 6 (1996) 925–942.
Ermakov, M.S., Minimax estimation in a deconvolution problem. J. Phys. A: Math. Gen. 25 (1992) 1273–1282. CrossRef
Ermakov, M.S., Asymptotically minimax and Bayes estimation in a deconvolution problem. Inverse Problems 19 (2003) 1339–1359. CrossRef
J. Fan, Asymptotic normality for deconvolution kernel estimators. Sankhia Ser. A 53 (1991) 97–110.
Fan, J., On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 (1991) 1257–1272. CrossRef
Goldenshluger, A., On pointwise adaptive nonparametric deconvolution. Bernoulli 5 (1999) 907–25. CrossRef
Yu, K. Golubev, B.Y. Levit, A.B. Tsybakov, Asymptotically efficient estimation of Analitic functions in Gaussian noise. Bernoulli 2 (1996) 167–181.
Ibragimov, I.A. and Hasminskii, R.Z., Estimation of distribution density belonging to a class of entire functions. Theory Probab. Appl. 27 (1982) 551–562. CrossRef
P.A. Jansson, Deconvolution, with application to Spectroscopy. New York, Academic (1984).
Johnstone, I.M., Kerkyacharian, G., Picard, D. and M.Raimondo, Wavelet deconvolution in a periodic setting. J. Roy. Stat. Soc. Ser B. 66 (2004) 547–573. CrossRef
Johnstone, I.M. and Raimondo, M., Periodic boxcar deconvolution and Diophantine approximation. Ann. Statist. 32 (2004) 1781–1805.
Kalifa, J. and Mallat, S., Threshholding estimators for linear inverse problems and deconvolutions. Ann. Stat. 31 (2003) 58–109.
Kassam, S. and Poor, H., Robust techniques for signal processing. A survey. Proc. IEEE 73 (1985) 433–481. CrossRef
M.R. Leadbetter, G. Lindgren and H. Rootzen, Extremes and Related Properties of Random sequences and Processes. Springer-Verlag NY (1986).
Neelamani, R., Choi, H., Baraniuk, R.G., ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems. IEEE Trans. Signal Process. 52 (2004) 418–433. CrossRef
Nussbaum, M., Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Stat. 24 (1996) 2399–2430.
Pensky, M. and Vidakovic, B., Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 (1999) 2033–2053.
Pinsker, M.S., Optimal filtration of square-integral signal in Gaussian noise. Problems Inform. Transm. 16 (1980) 52–68.
Schipper, M., Optimal rates and constants in L 2-minimax estimation of probability density functions. Math. Methods Stat. 5 (1996) 253–274.
Smola, A.J., Scholkopf, B. and Miller, K., The connection between regularization operators and support vector kernels. Newral Networks 11 (1998) 637–649. CrossRef
A. Tikhonov and V. Arsenin, Solution of Ill-Posed Problems. New-York, Wiley (1977).
A.B. Tsybakov, On the best rate of adaptive estimation in some inverse problems. C.R. Acad. Sci. Paris, Serie 1 330 (2000) 835–840.
N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series. New York, Wiley (1950).
* This paper was partially supported by RFFI Grants 02-01-00262, 4422.2006.1.