Robust and efficient recovery of a signal passed through a filter and then contaminated by non-Gaussian noise

Consider a channel where a continuous periodic input signal is passed through a linear filter and then is contaminated by an additive noise. The problem is to recover this signal when we observe n repeated realizations of the output signal. Adaptive efficient procedures, that are asymptotically minimax over all possible procedures, are known for channels with Gaussian noise and no filter (the case of direct observation). Efficient procedures, based on the smoothness of a recovered signal, are known for the case of Gaussian noise. Robust rate-optimal procedures are known as well. However, there are no results on robust and efficient data-driven procedures; moreover, the known results for the case of direct observation indicate that even a small deviation from Gaussian noise may lead to a drastic change. We show that for the considered case of indirect data and a particular class of so-called supersmooth filters there exists a procedure of recovery of an input signal that possesses the desired properties; namely, it is: adaptive to the smoothness of the input signal; robust to the distribution of the noise; globally and pointwise-efficient, that is, its minimax global and pointwise risks converge with the best constant and rate over all possible estimators as n→∞; and universal in the sense that for a wide class of linear (not necessarily bounded) operators the efficient estimator is a plug-in one. Furthermore, we explain how to employ the obtained asymptotic results for the practically important case of small n (large noise)