The Annals of Statistics

Modulation of estimators and confidence sets

Rudolf Beran and Lutz Dümbgen

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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 $\xi$.

Article information

Ann. Statist., Volume 26, Number 5 (1998), 1826-1856.

First available in Project Euclid: 21 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Adaptivity asymptotic minimax bootstrap bounded variation coverage probability isotonic regression orthogonal transformation signal recovery Stein’s unbiased estimator of risk tapering


Beran, Rudolf; Dümbgen, Lutz. Modulation of estimators and confidence sets. Ann. Statist. 26 (1998), no. 5, 1826--1856. doi:10.1214/aos/1024691359.

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