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Information bounds for inverse problems with application to deconvolution and Lévy models. (English. French summary) Zbl 1346.60063
Summary: 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.

60G51 Processes with independent increments; Lévy processes
62G20 Asymptotic properties of nonparametric inference
60H40 White noise theory
60J75 Jump processes (MSC2010)
62M05 Markov processes: estimation; hidden Markov models
62B15 Theory of statistical experiments
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