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.
@article{AIHPB_2015__51_4_1620_0, author = {Trabs, Mathias}, title = {Information bounds for inverse problems with application to deconvolution and {L\'evy} models}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {1620--1650}, publisher = {Gauthier-Villars}, volume = {51}, number = {4}, year = {2015}, doi = {10.1214/14-AIHP627}, mrnumber = {3414460}, zbl = {1346.60063}, language = {en}, url = {http://www.numdam.org/articles/10.1214/14-AIHP627/} }
TY - JOUR AU - Trabs, Mathias TI - Information bounds for inverse problems with application to deconvolution and Lévy models JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 1620 EP - 1650 VL - 51 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/14-AIHP627/ DO - 10.1214/14-AIHP627 LA - en ID - AIHPB_2015__51_4_1620_0 ER -
%0 Journal Article %A Trabs, Mathias %T Information bounds for inverse problems with application to deconvolution and Lévy models %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 1620-1650 %V 51 %N 4 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/14-AIHP627/ %R 10.1214/14-AIHP627 %G en %F AIHPB_2015__51_4_1620_0
Trabs, Mathias. Information bounds for inverse problems with application to deconvolution and Lévy models. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1620-1650. doi : 10.1214/14-AIHP627. http://www.numdam.org/articles/10.1214/14-AIHP627/
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