# Minimax and bayes estimation in deconvolution problem*

ESAIM: Probability and Statistics (2008)

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

## Access Full Article

top## Abstract

top## How to cite

topErmakov, 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

top- 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.
- C. Butucea, Deconvolution of supersmooth densities with smooth noise. Canad. J. Statist.32 (2004) 181–192. Zbl1056.62047
- C. Butucea and A.B. Tsybakov, Sharp optimality for density deconvolution with dominating bias. (2004), arXiv:math.ST/0409471. Zbl1141.62021
- 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
- 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.
- R.J. Carrol and P. Hall, Optimal rates of convergence for deconvolving a density J. Amer. Statist. Assoc.83 (1988) 1184–1186. Zbl0673.62033
- D.L. Donoho, Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal.2 (1992) 101–126. Zbl0826.65117
- S. Efroimovich, Nonparametric Curve Estimation: Methods, Theory and Applications. New York, Springer (1999).
- S. Efromovich and M. Pinsker, Sharp optimal and adaptive estimation for heteroscedastic nonparametric regression. Statistica Cinica6 (1996) 925–942. Zbl0857.62037
- M.S. Ermakov, Minimax estimation in a deconvolution problem. J. Phys. A: Math. Gen.25 (1992) 1273–1282. Zbl0765.62080
- M.S. Ermakov, Asymptotically minimax and Bayes estimation in a deconvolution problem. Inverse Problems19 (2003) 1339–1359. Zbl1040.62002
- J. Fan, Asymptotic normality for deconvolution kernel estimators. Sankhia Ser. A 53 (1991) 97–110. Zbl0729.62034
- J. Fan, On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist.19 (1991) 1257–1272. Zbl0729.62033
- A. Goldenshluger, On pointwise adaptive nonparametric deconvolution. Bernoulli5 (1999) 907–25. Zbl0953.62033
- 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
- 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
- P.A. Jansson, Deconvolution, with application to Spectroscopy. New York, Academic (1984).
- 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
- I.M. Johnstone and M. Raimondo, Periodic boxcar deconvolution and Diophantine approximation. Ann. Statist.32 (2004) 1781–1805. Zbl1056.62044
- J. Kalifa and S. Mallat, Threshholding estimators for linear inverse problems and deconvolutions. Ann. Stat.31 (2003) 58–109. Zbl1102.62318
- S. Kassam and H. Poor, Robust techniques for signal processing. A survey. Proc. IEEE73 (1985) 433–481. Zbl0569.62084
- M.R. Leadbetter, G. Lindgren and H. Rootzen, Extremes and Related Properties of Random sequences and Processes. Springer-Verlag NY (1986). Zbl0518.60021
- R. Neelamani, H. Choi, R.G. Baraniuk, ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems. IEEE Trans. Signal Process.52 (2004) 418–433.
- M. Nussbaum, Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Stat.24 (1996) 2399–2430. Zbl0867.62035
- M. Pensky and B. Vidakovic, Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist.27 (1999) 2033–2053. Zbl0962.62030
- M.S. Pinsker, Optimal filtration of square-integral signal in Gaussian noise. Problems Inform. Transm. 16 (1980) 52–68.
- M. Schipper, Optimal rates and constants in L2-minimax estimation of probability density functions. Math. Methods Stat.5 (1996) 253–274. Zbl0872.62043
- A.J. Smola, B. Scholkopf and K. Miller, The connection between regularization operators and support vector kernels. Newral Networks11 (1998) 637–649.
- A. Tikhonov and V. Arsenin, Solution of Ill-Posed Problems. New-York, Wiley (1977). Zbl0354.65028
- 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
- 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.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.