Modulation of estimators and confidence sets

Abstract

An unknown signal plus white noise is observed at $n$ discrete time points. Within a large convex class of linear estimators of $\xi$, we choose the estimator $\hat{\xi}$ that minimizes estimated quadratic risk. By construction, $\hat{\xi}$ 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 $\hat{\xi}$ 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 $\hat{\xi}$ and construct confidence sets for the unknown signal. These confidence sets are centered at $\hat{\xi}$, have correct asymptotic coverage probability and have relatively small risk as set-valued estimators of $\xi$.