Modulation Estimators and Con dence Sets

An unknown signal plus white noise is observed at n discrete time points. Within a large convex class of linear estimators of , we choose the estimator b that minimizes estimated quadratic risk. By construction, b is nonlinear. This estimation is done after orthogonal transformation of the data to a reasonable coordinate system. The procedure adaptively tapers the coeecients of the transformed data. If the class of candidate estimators satisses a uniform entropy condition, then b is asymptotically mini-max in Pinsker's sense over certain ellipsoids in the parameter space and shares one such asymptotic minimax property with the James-Stein estimator. We describe computational algorithms for b and construct conndence sets for the unknown signal. These conndence sets are centered at b , have correct asymptotic coverage probability, and have relatively small risk as set-valued estimators of .