Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Mathias Trabs

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 4 (2015), 1620-1650.

Dates
Received: 23 August 2013
Revised: 16 April 2014
Accepted: 6 May 2014
First available in Project Euclid: 21 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1445432055

Digital Object Identifier
doi:10.1214/14-AIHP627

Mathematical Reviews number (MathSciNet)
MR3414460

Zentralblatt MATH identifier
1346.60063

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60J75: Jump processes 62B15: Theory of statistical experiments 62G20: Asymptotic properties 62M05: Markov processes: estimation

Keywords
Convolution theorem Deconvolution Lévy process Nonlinear ill-posed inverse problem Semiparametric efficiency White noise model

Citation

Trabs, Mathias. Information bounds for inverse problems with application to deconvolution and Lévy models. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1620--1650. doi:10.1214/14-AIHP627. https://projecteuclid.org/euclid.aihp/1445432055


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