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

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.

Efficient nonparametric inference for discretely observed compound Poisson processes

“…This agrees with the observation in [36] that γ has no influence on the information lower bounds of the problem and also with the null limiting quantity for γ in Proposition 2.1. The latter and the h n / √ n-rate may seem surprising at first.…”

“…We identify their covariance structures and these coincide with the information-theoretic lower bounds developed by Trabs in [36], so our estimators are asymptotically efficient. The inverse nature of the problem translates into convoluted expressions for the covariances.…”

“…In this setting N should be estimated by λ F n and not by N n . This way we input our knowledge of λ and in view of the remarks we just made the estimator attains the information-theoretic lower bound developed in [36]. Clearly, N n would not attain it in this setting.…”

“…In [36], Trabs showed that if ν is absolutely continuous, the Cramér-Rao bound for the limiting variance of a 1/ √ n-consistent estimator of a functional R f dν is (2.16) with f 1 = f 2 = f 1 R \{0} . The truncation induced by the indicator is crucial: by basic Fourier analysis…”

“…Therefore, they estimate this generalised Lévy distribution in the norm L ∞ ((−∞, −ζ] ∪ [ζ, ∞)), for some ζ > 0 fixed. Their limiting Gaussian process is optimal as shown in [36], and their estimator is efficient. Their result applies to compound Poisson processes but does not address estimation around the origin.…”