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March, 1994 On Minimax Estimation of a Sparse Normal Mean Vector
Iain M. Johnstone
Ann. Statist. 22(1): 271-289 (March, 1994). DOI: 10.1214/aos/1176325368


Mallows has conjectured that among distributions which are Gaussian but for occasional contamination by additive noise, the one having least Fisher information has (two-sided) geometric contamination. A very similar problem arises in estimation of a nonnegative vector parameter in Gaussian white noise when it is known also that most [i.e., $(1 - \varepsilon)$] components are zero. We provide a partial asymptotic expansion of the minimax risk as $\varepsilon \rightarrow 0$. While the conjecture seems unlikely to be exactly true for finite $\varepsilon$, we verify it asymptotically up to the accuracy of the expansion. Numerical work suggests the expansion is accurate for $\varepsilon$ as large as 0.05. The best $l_1$-estimation rule is first- but not second-order minimax. The results bear on an earlier study of maximum entropy estimation and various questions in robustness and function estimation using wavelet bases.


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Iain M. Johnstone. "On Minimax Estimation of a Sparse Normal Mean Vector." Ann. Statist. 22 (1) 271 - 289, March, 1994.


Published: March, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0816.62007
MathSciNet: MR1272083
Digital Object Identifier: 10.1214/aos/1176325368

Primary: 62C20
Secondary: 62C10, 62G05

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.22 • No. 1 • March, 1994
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