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Beran, Rudolf; Dümbgen, Lutz

Modulation of estimators and confidence sets. (English) Zbl 1073.62538

Ann. Stat. 26, No. 5, 1826-1856 (1998).
Summary: 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 \(\xi\).

Cited in 1 Review
Cited in 33 Documents

MSC:

62H12 Estimation in multivariate analysis
62B10 Statistical aspects of information-theoretic topics
62F25 Parametric tolerance and confidence regions

Keywords:

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

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