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October 1998 Modulation of estimators and confidence sets
Rudolf Beran, Lutz Dümbgen
Ann. Statist. 26(5): 1826-1856 (October 1998). DOI: 10.1214/aos/1024691359

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 $\xi$.

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Rudolf Beran. Lutz Dümbgen. "Modulation of estimators and confidence sets." Ann. Statist. 26 (5) 1826 - 1856, October 1998. https://doi.org/10.1214/aos/1024691359

Information

Published: October 1998
First available in Project Euclid: 21 June 2002

zbMATH: 1073.62538
MathSciNet: MR1673280
Digital Object Identifier: 10.1214/aos/1024691359

Subjects:
Primary: 62H12
Secondary: 62M10

Rights: Copyright © 1998 Institute of Mathematical Statistics

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Vol.26 • No. 5 • October 1998
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