Modulation of estimators and confidence sets

An unknown signal plus white noise is observed at n discrete time points. W ithin a large convex class of linear estimators of xi, we choose the estima tor <(xi)over cap> that minimizes estimated quadratic risk. By construction , <(xi)over cap> is nonlinear. This estimation is done after orthogonal tra nsformation of the data to a reasonable coordinate system. The procedure ad aptively tapers the coefficients of the transformed data. If the class of c andidate estimators satisfies a uniform entropy condition, then <(xi)over c ap> is asymptotically minimax in Pinsker's sense over certain ellipsoids in the parameter space and shares one such asymptotic minimax property with t he James-Stein estimator. We describe computational algorithms for <(xi)ove r cap> and construct confidence sets for the unknown signal. These confiden ce sets are centered at <(xi)over cap>, have correct asymptotic coverage pr obability and have relatively small risk as set-valued estimators of xi.