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On convergence rates equivalency and sampling strategies in functional deconvolution models. (English) Zbl 1352.62050
Summary: Using the asymptotical minimax framework, we examine convergence rates equivalency between a continuous functional deconvolution model and its real-life discrete counterpart over a wide range of Besov balls and for the \(L^{2}\)-risk. For this purpose, all possible models are divided into three groups. For the models in the first group, which we call uniform, the convergence rates in the discrete and the continuous models coincide no matter what the sampling scheme is chosen, and hence the replacement of the discrete model by its continuous counterpart is legitimate. For the models in the second group, to which we refer as regular, one can point out the best sampling strategy in the discrete model, but not every sampling scheme leads to the same convergence rates; there are at least two sampling schemes which deliver different convergence rates in the discrete model (i.e., at least one of the discrete models leads to convergence rates that are different from the convergence rates in the continuous model). The third group consists of models for which, in general, it is impossible to devise the best sampling strategy; we call these models irregular.
We formulate the conditions when each of these situations takes place. In the regular case, we not only point out the number and the selection of sampling points which deliver the fastest convergence rates in the discrete model but also investigate when, in the case of an arbitrary sampling scheme, the convergence rates in the continuous model coincide or do not coincide with the convergence rates in the discrete model. We also study what happens if one chooses a uniform, or a more general pseudo-uniform, sampling scheme which can be viewed as an intuitive replacement of the continuous model. Finally, as a representative of the irregular case, we study functional deconvolution with a boxcar-like blurring function since this model has a number of important applications. All theoretical results presented in the paper are illustrated by numerous examples; many of which are motivated directly by a multitude of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation. The theoretical performance of the suggested estimator in the multichannel deconvolution model with a boxcar-like blurring function is also supplemented by a limited simulation study and compared to an estimator available in the current literature. The paper concludes that in both regular and irregular cases one should be extremely careful when replacing a discrete functional deconvolution model by its continuous counterpart.

MSC:
62G05 Nonparametric estimation
62G08 Nonparametric regression and quantile regression
Software:
ForWaRD; reccv
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[1] Abramovich, F. and Silverman, B. W. (1998). Wavelet decomposition approaches to statistical inverse problems. Biometrika 85 115-129. JSTOR: · Zbl 0908.62095 · doi:10.1093/biomet/85.1.115 · www3.oup.co.uk
[2] Antoniadis, A., Bigot, J. and Sapatinas, T. (2001). Wavelet estimators in nonparametric regression: A comparative simulation study. Journal of Statistical Software 6 Article 6.
[3] Bender, C. M. and Orzag, S. A. (1978). Advanced Mathematical Methods for Scientists and Engineers . McGraw-Hill, New York. · Zbl 0417.34001
[4] Brown, L. D., Cai, T., Low, M. G. and Zhang, C.-H. (2002). Asymptotic equivalence theory for nonparametric regression with random design. Ann. Statist. 30 688-707. · Zbl 1029.62044 · doi:10.1214/aos/1028674838 · euclid:aos/1028674838
[5] Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384-2398. · Zbl 0867.62022 · doi:10.1214/aos/1032181159
[6] Casey, S. D. and Walnut, D. F. (1994). Systems of convolution equations, deconvolution, Shannon sampling, and the wavelet and Gabor transforms. SIAM Rev. 36 537-577. JSTOR: · Zbl 0814.45001 · doi:10.1137/1036140 · links.jstor.org
[7] Cavalier, L. and Raimondo, M. (2007). Wavelet deconvolution with noisy eigenvalues. IEEE Trans. Signal Process. 55 2414-2424. · Zbl 1391.94160 · doi:10.1109/TSP.2007.893754
[8] Chesneau, C. (2008). Wavelet estimation via block thresholding: A minimax study under L p -risk. Statist. Sinica 18 1007-1024. · Zbl 05361942
[9] Cirelson, B. S., Ibragimov, I. A. and Sudakov, V. N. (1976). Norm of Gaussian sample function. In Proceedings of the 3rd Japan-U.S.S.R. Symposium on Probability Theory. Lecture Notes in Math. 550 20-41. Springer, Berlin. · Zbl 0359.60019
[10] De Canditiis, D. and Pensky, M. (2004). Discussion on the meeting on “Statistical Approaches to Inverse Problems.” J. R. Stat. Soc. Ser. B Stat. Methodol. 66 638-640.
[11] De Canditiis, D. and Pensky, M. (2006). Simultaneous wavelet deconvolution in periodic setting. Scand. J. Statist. 33 293-306. · Zbl 1124.62019 · doi:10.1111/j.1467-9469.2006.00463.x
[12] Donoho, D. L. (1995). Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 101-126. · Zbl 0826.65117 · doi:10.1006/acha.1995.1008
[13] Donoho, D. L. and Raimondo, M. (2004). Translation invariant deconvolution in a periodic setting. Int. J. Wavelets Multiresolut. Inf. Process. 14 415-432. · Zbl 1071.62088 · doi:10.1142/S0219691304000640
[14] Golubev, G. (2004). The principle of penalized empirical risk in severely ill-posed problems. Probab. Theory Related Fields 130 18-38. · Zbl 1064.62011 · doi:10.1007/s00440-004-0362-y
[15] Golubev, G. K. and Khasminskii, R. Z. (1999). A statistical approach to some inverse problems for partial differential equations. Probl. Inf. Transm. 35 136-149. · Zbl 0947.35174
[16] Gradshtein, I. S. and Ryzhik, I. M. (1980). Tables of Integrals, Series, and Products . Academic Press, New York.
[17] Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statistics 129 . Springer, New York. · Zbl 0899.62002
[18] Harsdorf, S. and Reuter, R. (2000). Stable deconvolution of noisy lidar signals. In Proceedings of EARSeL-SIG-Workshop LIDAR 16-17. Dresden/FRG.
[19] Johnstone, I. M. (2002). Function Estimation in Gaussian Noise: Sequence Models . Unpublished Monograph. Available at http://www-stat.stanford.edu/ imj/.
[20] Johnstone, I. M., Kerkyacharian, G., Picard, D. and Raimondo, M. (2004). Wavelet deconvolution in a periodic setting (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 66 547-573. JSTOR: · Zbl 1046.62039 · doi:10.1111/j.1467-9868.2004.02056.x · links.jstor.org
[21] Johnstone, I. M. and Raimondo, M. (2004). Periodic boxcar deconvolution and Diophantine approximation. Ann. Statist. 32 1781-1804. · Zbl 1056.62044 · doi:10.1214/009053604000000391
[22] Kalifa, J. and Mallat, S. (2003). Thresholding estimators for linear inverse problems and deconvolutions. Ann. Statist. 31 58-109. · Zbl 1102.62318 · doi:10.1214/aos/1046294458
[23] Kerkyacharian, G., Picard, D. and Raimondo, M. (2007). Adaptive boxcar deconvolution on full Lebesgue measure sets. Statist. Sinica 7 317-340. · Zbl 1145.62066
[24] Kolaczyk, E. D. (1994). Wavelet methods for the inversion of certain homogeneous linear operators in the presence of noisy data. Ph.D. dissertation, Dept. Statistics, Stanford Univ.
[25] Lang, S. (1966). Introduction to Diophantine Approximations . Springer, New York. · Zbl 0144.04005
[26] Lattes, R. and Lions, J. L. (1967). Methode de Quasi-Reversibilite et Applications. Travoux et Recherche Mathematiques 15 . Dunod, Paris. · Zbl 0159.20803
[27] Meyer, Y. (1992). Wavelets and Operators . Cambridge Univ. Press, Cambridge. · Zbl 0776.42019
[28] Müller, H. G. and Stadmüller, U. (1987). Variable bandwidth kernel estimators of regression curves. Ann. Statist. 15 182-201. · Zbl 0634.62032 · doi:10.1214/aos/1176350260
[29] Neelamani, R., Choi, H. and Baraniuk, R. (2004). Forward: Fourier-wavelet regularized deconvolution for ill-conditioned systems. IEEE Trans. Signal Process. 52 418-433. · Zbl 1369.94238 · doi:10.1109/TSP.2003.821103
[30] Park, Y. J., Dho, S. W. and Kong, H. J. (1997). Deconvolution of long-pulse lidar signals with matrix formulation. Appl. Optics 36 5158-5161.
[31] Pensky, M. and Sapatinas, T. (2009a). Functional deconvolution in a periodic case: Uniform case. Ann. Statist. 37 73-104. · Zbl 1274.62253 · doi:10.1214/07-AOS552
[32] Pensky, M. and Sapatinas, T. (2009b). Diophantine approximation and the problem of estimation of the initial speed of a wave on a finite interval in a stochastic setting. Technical Report TR-07-2009, Dept. Mathematics and Statistics, Univ. Cyprus. · Zbl 1274.62253
[33] Petsa, A. and Sapatinas, T. (2009). Minimax convergence rates under the L p -risk in the functional deconvolution model. Statist. Probab. Lett. 79 1568-1576. [Erratum: Statist. Probab. Lett. 79 1890 (2009).] · Zbl 1165.62320 · doi:10.1016/j.spl.2009.03.028
[34] Reiss, M. (2008). Asymptotic equivalence for nonparametric regression with multivariate and random design. Ann. Statist. 36 1957-1982. · Zbl 1142.62023 · doi:10.1214/07-AOS525
[35] Schmidt, W. (1980). Diophantine Approximation. Lecture Notes in Math. 785 . Springer, Berlin. · Zbl 0421.10019
[36] Willer, T. (2005). Deconvolution in white noise with a random blurring function. Preprint. Avaiable at
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