Modulation of estimators and confidence sets

An unknown signal plus white noise is observed at n discrete time points. Within a large convex class of linear estimators of ξ, we choose the estimator ξ̂ that minimizes estimated quadratic risk. By construction, ξ̂ is nonlinear. This estimation is done after orthogonal transformation of the data to a reasonable coordinate system. The procedure adaptively tapers the coefficients of the transformed data. If the class of candidate estimators satisfies a uniform entropy condition, then ξ̂ is asymptotically minimax 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 ξ̂ and construct confidence sets for the unknown signal. These confidence sets are centered at ξ̂, have correct asymptotic coverage probability and have relatively small risk as set-valued estimators of ξ.