ROBUST AND EFFICIENT RECOVERY OF A SIGNAL PASSED THROUGH A FILTER ANDTHEN 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 real izations of the output signal, Adaptive efficient procedures, that are asymptotically minimax ol-er all possible procedures, are known for t he channels with Gaussian noise and no filter (the case of direct obse rvation), Efficient procedures, based on smoothness of a recovered sig nal, are known for the case of Gaussian noise, Robust rate-optimal pro cedures are known as well. However, there is no results on robust and efficient data-driven procedures; moreover, the known results for the case of direct observation indicate that even a smalt deviation from G aussian noise may lead to a drastic change, We show that for the consi dered case of indirect data and a particular class of so-called supers mooth filters there exists a procedure of recovery of an input signal that possesses the desired properties; namely, it is: adaptive to smoo thness of input signal; robust to the distribution of a noise; globall y and pointwise-efficient, that is, its minimax global and pointwise r isks converge with the best constant and rate over all possible estima tors as n --> infinity; universal in the sense that for a wide class o f linear (not necessarily bounded) operators the efficient estimator i s a plug-in one. Furthermore, we explain how to employ the obtained as ymptotic results for the practically important case of small n (large noise).