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Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 3, Pages 534–556 (Mi tvp269)  

This article is cited in 16 scientific papers (total in 16 papers)

Block thresholding and sharp adaptive estimation in severely ill-posed inverse problems

L. Cavalier, Yu. F. Golubev, O. V. Lepskiĭ, A. Tsybakov

Université de Provence

Abstract: We consider the problem of solving linear operator equations from noisy data under the assumptions that the singular values of the operator decrease exponentially fast and that the underlying solution is also exponentially smooth in the Fourier domain. We suggest an estimator of the solution based on a running version of block thresholding in the space of Fourier coefficients. This estimator is shown to be sharp adaptive to the unknown smoothness of the solution.

Keywords: linear operator equation, white Gaussian noise, adaptive estimation, running block thresholding.


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English version:
Theory of Probability and its Applications, 2004, 48:3, 426–446

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Received: 23.07.2002

Citation: L. Cavalier, Yu. F. Golubev, O. V. Lepskiǐ, A. Tsybakov, “Block thresholding and sharp adaptive estimation in severely ill-posed inverse problems”, Teor. Veroyatnost. i Primenen., 48:3 (2003), 534–556; Theory Probab. Appl., 48:3 (2004), 426–446

Citation in format AMSBIB
\by L.~Cavalier, Yu.~F.~Golubev, O.~V.~Lepski{\v\i}, A.~Tsybakov
\paper Block thresholding and sharp adaptive estimation in severely ill-posed inverse problems
\jour Teor. Veroyatnost. i Primenen.
\yr 2003
\vol 48
\issue 3
\pages 534--556
\jour Theory Probab. Appl.
\yr 2004
\vol 48
\issue 3
\pages 426--446

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    1. G. K. Golubev, “Method of Risk Envelopes in Estimation of Linear Functionals”, Problems Inform. Transmission, 40:1 (2004), 53–65  mathnet  crossref  mathscinet  zmath
    2. Cavalier L., Golubev Yu., “Risk hull method and regularization by projections of ill–posed inverse problems”, Annals of Statistics, 34:4 (2006), 1653–1677  crossref  mathscinet  zmath  isi  scopus
    3. Comte F., Rozenholc Y., Taupin M.-L., “Penalized contrast estimator for adaptive density deconvolution”, Canadian Journal of Statistics–Revue Canadienne de Statistique, 34:3 (2006), 431–452  crossref  mathscinet  zmath  isi  scopus
    4. Theory Probab. Appl., 52:1 (2008), 24–39  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Ngoc Thanh Mai Pham, “A statistical minimax approach to the Hausdorff moment problem”, Inverse Problems, 24:4 (2008), 045018  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. Cavalier L., “Nonparametric statistical inverse problems”, Inverse Problems, 24:3 (2008), 034004  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. Hengartner N.W., Takala B.E., Michalak S.E., Wender S.A., “Evaluating experiments for estimating the bit failure cross-section of semiconductors using a colored spectrum neutron beam”, Technometrics, 50:1 (2008), 8–14  crossref  mathscinet  isi  scopus
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    10. Butucea C., Meziani K., “Quadratic Functional Estimation in Inverse Problems”, Stat. Methodol., 8:1, SI (2011), 31–41  crossref  mathscinet  zmath  isi  scopus
    11. Ingster Yu.I., Sapatinas T., Suslina I.A., “Minimax Signal Detection in Ill-Posed Inverse Problems”, Ann. Stat., 40:3 (2012), 1524–1549  crossref  mathscinet  zmath  isi  elib  scopus
    12. Cai T.T., “Minimax and Adaptive Inference in Nonparametric Function Estimation”, Stat. Sci., 27:1 (2012), 31–50  crossref  mathscinet  zmath  isi  scopus
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    14. Paul D., Peng J., Burman P., “Nonparametric estimation of dynamics of monotone trajectories”, Ann. Stat., 44:6 (2016), 2401–2432  crossref  mathscinet  zmath  isi  scopus
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    16. Werner F., “Adaptivity and Oracle Inequalities in Linear Statistical Inverse Problems: a (Numerical) Survey”, New Trends in Parameter Identification For Mathematical Models, Trends in Mathematics, eds. Hofmann B., Leitao A., Zubelli J., Birkhauser Verlag Ag, 2018, 291–316  crossref  mathscinet  zmath  isi  scopus
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