Information bounds for inverse problems with application to deconvolution and Lévy models

Trabs, Mathias

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1620-1650.

Si une fonctionnelle dans un problème inverse non-paramétrique peut être estimée à vitesse paramétrique, alors la vitesse minimax ne donne aucune information sur le caractère mal posé du problème. Pour avoir une borne inférieure plus précise, nous étudions l’efficacité semi-paramétrique dans le sens de Hájek–Le Cam pour l’estimation fonctionnelle dans des modèles indirects réguliers. Ces derniers sont caractérisés comme modèles que l’on peut approcher localement par un modèle linéaire de bruit blanc décrit par l’opérateur de score généralisé. Un théorème de convolution pour des modèles indirects réguliers est prouvé. Ceci s’applique à une large classe de problèmes statistiques inverses, comme montré pour les modèles prototypes du bruit blanc et de la déconvolution. Il est spécialement utile pour des modèles non-linéaires. Nous discutons en détails un modèle non-linéaire de déconvolution où un processus de Lévy est observé à basse fréquence, en obtenant une borne d’information pour l’estimation de fonctionnelles linéaires de la mesure de sauts.

If a functional in a nonparametric inverse problem can be estimated with parametric rate, then the minimax rate gives no information about the ill-posedness of the problem. To have a more precise lower bound, we study semiparametric efficiency in the sense of Hájek–Le Cam for functional estimation in regular indirect models. These are characterized as models that can be locally approximated by a linear white noise model that is described by the generalized score operator. A convolution theorem for regular indirect models is proved. This applies to a large class of statistical inverse problems, which is illustrated for the prototypical white noise and deconvolution model. It is especially useful for nonlinear models. We discuss in detail a nonlinear model of deconvolution type where a Lévy process is observed at low frequency, concluding an information bound for the estimation of linear functionals of the jump measure.

Mots clés : convolution theorem, deconvolution, lévy process, nonlinear ill-posed inverse problem, semiparametric efficiency, White noise model

[1] D. Belomestny and M. Reiß. Spectral calibration of exponential Lévy models. Finance Stoch. 10 (4) (2006) 449–474. | MR | Zbl

[2] P. J. Bickel, C. A. J. Klaassen, Y. Ritov and J. A. Wellner. Efficient and Adaptive Estimation for Semiparametric Models. Springer, New York, 1998. | MR | Zbl

[3] N. Bissantz, T. Hohage and A. Munk. Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise. Inverse Probl. 20 (6) (2004) 1773–1789. | MR | Zbl

[4] L. D. Brown and M. G. Low. Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 (6) (1996) 2384–2398. | MR | Zbl

[5] B. Buchmann and R. Grübel. Decompounding: An estimation problem for Poisson random sums. Ann. Statist. 31 (4) (2003) 1054–1074. | MR | Zbl

[6] L. Cavalier. Nonparametric statistical inverse problems. Inverse Probl. 24 (3) 034004 (2008). | MR | Zbl

[7] L. Cavalier, G. K. Golubev, D. Picard and A. B. Tsybakov. Oracle inequalities for inverse problems. Ann. Statist. 30 (3) (2002) 843–874. | MR | Zbl

[8] E. Clément, S. Delattre and A. Gloter. An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility. Stochastic Process. Appl. 123 (7) (2013) 2500–2521. | MR | Zbl

[9] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge, 1992. | DOI | MR | Zbl

[10] H. W. Engl, M. Hanke and A. Neubauer. Regularization of Inverse Problems. Mathematics and Its Applications 375. Kluwer Academic, Dordrecht, 1996. | MR | Zbl

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

[12] E. Giné and R. Nickl. Uniform central limit theorems for kernel density estimators. Probab. Theory Related Fields 141 (3–4) (2008) 333–387. | MR | Zbl

[13] A. Goldenshluger and S. V. Pereverzev. Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations. Probab. Theory Related Fields 118 (2) (2000) 169–186. | MR | Zbl

[14] A. Goldenshluger and S. V. Pereverzev. On adaptive inverse estimation of linear functionals in Hilbert scales. Bernoulli 9 (5) (2003) 783–807. | MR | Zbl

[15] J. Jacod. Une application de la topologie d’Emery: le processus information d’un modèle statistique filtré. In Séminaire de Probabilités, XXIII 448–474. Lecture Notes in Math. 1372. Springer, Berlin, 1989. | Numdam | MR | Zbl

[16] J. Jacod. Regularity, partial regularity, partial information process, for a filtered statistical model. Probab. Theory Related Fields 86 (3) (1990) 305–335. | MR | Zbl

[17] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin, 2003. | MR | Zbl

[18] A. Khoujmane, F. Ruymgaart and M. Shubov. The convolution theorem for estimating linear functionals in indirect nonparametric regression models. J. Statist. Plann. Inference 137 (3) (2007) 811–820. | MR | Zbl

[19] C. A. J. Klaassen, E.-J. Lee and F. H. Ruymgaart. On efficiency of indirect estimation of nonparametric regression functions. In Algebraic Methods in Statistics and Probability (Notre Dame, IN, 2000) 173–184. Contemp. Math. 287. Amer. Math. Soc., Providence, RI, 2001. | MR | Zbl

[20] C. A. J. Klaassen, E.-J. Lee and F. H. Ruymgaart. Asymptotically efficient estimation of linear functionals in inverse regression models. J. Nonparametr. Stat. 17 (7) (2005) 819–831. | MR | Zbl

[21] H. H. Kuo. Gaussian Measures in Banach Spaces. Lecture Notes in Math. 463. Springer, Berlin, 1975. | MR | Zbl

[22] L. Le Cam. Limits of experiments. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. I: Theory of Statistics 245–261. Univ. California Press, Berkeley, CA, 1972. | MR | Zbl

[23] F. Liese. Estimates of Hellinger integrals of infinitely divisible distributions. Kybernetika (Prague) 23 (3) (1987) 227–238. | MR | Zbl

[24] A. Meister. Deconvolution Problems in Nonparametric Statistics. Lecture Notes in Statist. Springer, Berlin, 2009. | DOI | MR | Zbl

[25] F. Natterer. Error bounds for Tikhonov regularization in Hilbert scales. Appl. Anal. 18 (1–2) (1984) 29–37. | MR | Zbl

[26] M. H. Neumann and M. Reiß. Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15 (1) (2009) 223–248. | MR | Zbl

[27] R. Nickl and M. Reiß. A Donsker theorem for Lévy measures. J. Funct. Anal. 263 (10) (2012) 3306–3332. | MR | Zbl

[28] M. Nussbaum. Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 (6) (1996) 2399–2430. | MR | Zbl

[29] M. Reiß. Testing the characteristics of a Lévy process. Stochastic Process. Appl. 123 (7) (2013) 2808–2828. | MR | Zbl

[30] K.-I. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge, 1999. | MR | Zbl

[31] J. Söhl and M. Trabs. A uniform central limit theorem and efficiency for deconvolution estimators. Electron. J. Stat. 6 (2012) 2486–2518. | DOI | MR | Zbl

[32] M. Trabs. Calibration of self-decomposable Lévy models. Bernoulli 20 (1) (2014) 109–140. | MR | Zbl

[33] A. W. Van Der Vaart. Statistical Estimation in Large Parameter Spaces. CWI Tract 44. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1988. | MR | Zbl

[34] A. W. Van Der Vaart. On differentiable functionals. Ann. Statist. 19 (1) (1991) 178–204. | MR | Zbl

[35] A. W. Van Der Vaart. Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press, Cambridge, 1998. | MR | Zbl

[36] A. W. Van Der Vaart and J. A. Wellner. Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York, 1996. | DOI | MR | Zbl

[37] A. C. M. Van Rooij, F. H. Ruymgaart and W. R. Van Zwet. Asymptotic efficiency of inverse estimators. Teor. Veroyatn. Primen. 44 (4) (1999) 826–844. Translation in Theory Probab. Appl. 44 (4) 722–738. | MR | Zbl

[38] H. Witting. Mathematische Statistik. I: Parametrische Verfahren bei festem Stichprobenumfang. B. G. Teubner, Stuttgart, 1985. | DOI | MR | Zbl