Anisotropic adaptive kernel deconvolution

Comte, F.; Lacour, C.

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 569-609.

Dans ce travail, nous considérons un modèle de convolution multidimensionnel, pour lequel nous proposons des estimateurs à noyau anisotropes pour reconstruire la densité $f$ d’un signal mesuré avec un bruit additif. Pour ce faire, nous généralisons les estimateurs de Fan (Ann. Statist. 19(3) (1991) 1257-1272) à un contexte multidimensionnel et nous appliquons une méthode de sélection de fenêtre dans l'esprit des idées récentes développées par Goldenshluger et Lepski (Ann. Statist. 39(3) (2011) 1608-1632) pour l’estimation de densité en l’absence de bruit. Nous considérons tout d’abord le problème de l’estimation ponctuelle, et nous étudions ensuite le risque global intégré. Nos estimateurs dépendent d’une fenêtre aléatoire sélectionnée de façon automatique. Nous considérons les cas où les composantes du bruit, supposées connues, peuvent être ordinairement ou super régulières. De plus, nous étudions des classes de fonctions $f$ à estimer aussi bien dans des espaces de Hölder anisotropes que dans des espaces de Sobolev. Nous prouvons des bornes de risque non asymptotiques ainsi que des vitesses de convergence asymptotiques pour nos estimateurs adaptatifs, en même temps que des bornes inférieures dans un grand nombre de cas. Des simulations illustrent la méthode en s’appuyant sur des algorithmes de transformation de Fourier rapide. En conclusion, nous proposons une extension de la méthode lorsque la loi du bruit n’est plus connue, mais remplacée par un échantillon préliminaire où le bruit seul est observé.

In this paper, we consider a multidimensional convolution model for which we provide adaptive anisotropic kernel estimators of a signal density $f$ measured with additive error. For this, we generalize Fan’s (Ann. Statist. 19(3) (1991) 1257-1272) estimators to multidimensional setting and use a bandwidth selection device in the spirit of Goldenshluger and Lepski's (Ann. Statist. 39(3) (2011) 1608-1632) proposal for density estimation without noise. We consider first the pointwise setting and then, we study the integrated risk. Our estimators depend on an automatically selected random bandwidth. We assume both ordinary and super smooth components for measurement errors, which have known density. We also consider both anisotropic Hölder and Sobolev classes for $f$. We provide nonasymptotic risk bounds and asymptotic rates for the resulting data driven estimator, together with lower bounds in most cases. We provide an illustrative simulation study, involving the use of Fast Fourier Transform algorithms. We conclude by a proposal of extension of the method to the case of unknown noise density, when a preliminary pure noise sample is available.

[1] L. Birgé and P. Massart. Minimum contrast estimators on sieves: Exponential bounds and rates of convergence. Bernoulli 4(3) (1998) 329-375. | MR | Zbl

[2] C. Butucea. Deconvolution of supersmooth densities with smooth noise. Canad. J. Statist. 32(2) (2004) 181-192. | MR | Zbl

[3] C. Butucea and F. Comte. Adaptive estimation of linear functionals in the convolution model and applications. Bernoulli 15(1) (2009) 69-98. DOI:10.3150/08-BEJ146. Available at http://dx.doi.org/10.3150/08-BEJ146. | MR | Zbl

[4] C. Butucea and A. B. Tsybakov. Sharp optimality in density deconvolution with dominating bias. I. Theory Probab. Appl. 52(1) (2008) 24-39. DOI:10.1137/S0040585X97982840. Available at http://dx.doi.org/10.1137/S0040585X97982840. | MR | Zbl

[5] C. Butucea and A. B. Tsybakov. Sharp optimality in density deconvolution with dominating bias. II. Theory Probab. Appl. 52(2) (2008) 237-249. | MR | Zbl

[6] F. Comte and C. Lacour. Data-driven density estimation in the presence of additive noise with unknown distribution. J. R. Stat. Soc. Ser. B Stat. Methodol. 73(4) (2011) 601-627. DOI:10.1111/j.1467-9868.2011.00775.x. Available at http://dx.doi.org/10.1111/j.1467-9868.2011.00775.x. | MR | Zbl

[7] F. Comte and T. Rebafka. Adaptive density estimation in the pile-up model involving measurement errors. Preprint MAP5 2010-32, 2010. | MR

[8] F. Comte, Y. Rozenholc and M.-L. Taupin. Penalized contrast estimator for adaptive density deconvolution. Canad. J. Statist. 34(3) (2006) 431-452. | MR | Zbl

[9] A. Delaigle and I. Gijbels. Bootstrap bandwidth selection in kernel density estimation from a contaminated sample. Ann. Inst. Statist. Math. 56(1) (2004) 19-47. | MR | Zbl

[10] A. Delaigle, P. Hall and A. Meister. On deconvolution with repeated measurements. Ann. Statist. 36(2) (2008) 665-685. DOI:10.1214/009053607000000884. Available at http://dx.doi.org/10.1214/009053607000000884. | MR | Zbl

[11] L. Devroye. Nonuniform Random Variate Generation. Springer, New York, 1986. | MR | Zbl

[12] L. Devroye. The double kernel method in density estimation. Ann. Inst. H. Poincaré Probab. Stat. 25(4) (1989) 533-580. | Numdam | MR | Zbl

[13] M. Doumic, M. Hoffmann, P. Reynaud-Bouret and V. Rivoirard. Nonparametric estimation of the division rate of a size-structured population. Working paper, 2011. Available at http://hal.archives-ouvertes.fr/hal-00578694/fr/. | MR

[14] J. Fan. On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19(3) (1991) 1257-1272. | MR | Zbl

[15] J. Fan. Adaptively local one-dimensional subproblems with application to a deconvolution problem. Ann. Statist. 21(2) (1993) 600-610. DOI:10.1214/aos/1176349139. Available at http://dx.doi.org/10.1214/aos/1176349139. | MR | Zbl

[16] A. Goldenshluger and O. Lepski. Uniform bounds for norms of sums of independent random functions. Ann. Probab. 39(6) (2011) 2318-2384. DOI:10.1214/10-AOP595. Available at http://dx.doi.org/10.1214/10-AOP595. | MR | Zbl

[17] A. Goldenshluger and O. Lepski. Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality. Ann. Statist. 39(3) (2011) 1608-1632. | MR | Zbl

[18] P. Hall and A. Meister. A ridge-parameter approach to deconvolution. Ann. Statist. 35(4) (2007) 1535-1558. DOI:10.1214/009053607000000028. Available at http://dx.doi.org/10.1214/009053607000000028. | MR | Zbl

[19] J. Johannes. Deconvolution with unknown error distribution. Ann. Statist. 37(5A) (2009) 2301-2323. DOI:10.1214/08-AOS652. Available at http://dx.doi.org/10.1214/08-AOS652. | MR | Zbl

[20] G. Kerkyacharian, O. Lepski and D. Picard. Nonlinear estimation in anisotropic multi-index denoising. Probab. Theory Related Fields 121 (2001) 137-170. | MR | Zbl

[21] T. Klein and E. Rio. Concentration around the mean for maxima of empirical processes. Ann. Probab. 33(3) (2005) 1060-1077. DOI:10.1214/009117905000000044. Available at http://dx.doi.org/10.1214/009117905000000044. | MR | Zbl

[22] C. Lacour. Rates of convergence for nonparametric deconvolution. C. R. Math. Acad. Sci. Paris 342(11) (2006) 877-882. DOI:10.1016/j.crma.2006.04.006. Available at http://dx.doi.org/10.1016/j.crma.2006.04.006. | MR | Zbl

[23] E. Masry. Multivariate probability density deconvolution for stationary random processes. IEEE Trans. Inform. Theory 37(4) (1991) 1105-1115. DOI:10.1109/18.87002. Available at http://dx.doi.org/10.1109/18.87002. | MR | Zbl

[24] A. Meister. Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions. Inverse Problems 24(1) (2008) 015003. DOI:10.1088/0266-5611/24/1/015003. | MR | Zbl

[25] A. Meister. Deconvolution Problems in Nonparametric Statistics. Lecture Notes in Statistics 193. Springer, Berlin, 2009. | MR | Zbl

[26] M. H. Neumann. On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Stat. 7(4) (1997) 307-330. DOI:10.1080/10485259708832708. Available at http://dx.doi.org/10.1080/10485259708832708. | MR | Zbl

[27] S. M. Nikol'Skiĭ. Approximation of Functions of Several Variables and Imbedding Theorems. Springer, New York, 1975. Translated from the Russian by John M. Danskin, Jr., Die Grundlehren der Mathematischen Wissenschaften, Band 205. | MR | Zbl

[28] M. Pensky and B. Vidakovic. Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27(6) (1999) 2033-2053. | MR | Zbl

[29] H. Triebel. Theory of Function Spaces. III. Monographs in Mathematics 100. Birkhäuser, Basel, 2006. | MR | Zbl

[30] A. B. Tsybakov. Introduction to Nonparametric Estimation. Springer, New York, 2009. | MR | Zbl

[31] E. Youndjé and M. T. Wells. Optimal bandwidth selection for multivariate kernel deconvolution density estimation. TEST 17(1) (2008) 138-162. DOI:10.1007/s11749-006-0027-5. Available at http://dx.doi.org/10.1007/s11749-006-0027-5. | MR | Zbl