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March 1996 Neo-classical minimax problems, thresholding and adaptive function estimation
David L. Donoho, Iain M. Johnstone
Bernoulli 2(1): 39-62 (March 1996). DOI: 10.3150/bj/1193758789


We study the problem of estimating θ from data Y ~ N(θ,σ2) under squared-error loss. We define three new scalar minimax problems in which the risk is weighted by the size of θ. Simple thresholding gives asymptotically minimax estimates in all three problems. We indicate the relationships of the new problems to each other and to two other neo-classical problems: the problems of the bounded normal mean and of the risk-constrained normal mean. Via the wavelet transform, these results have implications for adaptive function estimation in two settings: estimating functions of unknown type and degree of smoothness in a global 2 norm; and estimating a function of unknown degree of local Hölder smoothness at a fixed point. In the latter setting, the scalar minimax results imply: Lepskii's results that it is not possible fully to adapt the unknown degree of smoothness without incurring a performance cost; and that simple thresholding of the empirical wavelet transform gives an estimate of a function at a fixed point which is, to within constants, optimally adaptive to unknown degree of smoothness.


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David L. Donoho. Iain M. Johnstone. "Neo-classical minimax problems, thresholding and adaptive function estimation." Bernoulli 2 (1) 39 - 62, March 1996.


Published: March 1996
First available in Project Euclid: 30 October 2007

MathSciNet: MR1394051
zbMATH: 0877.62035
Digital Object Identifier: 10.3150/bj/1193758789

Keywords: adaptive estimation , ℓ^p balls , minimax estimation , weak ℓ^p balls

Rights: Copyright © 1996 Bernoulli Society for Mathematical Statistics and Probability


Vol.2 • No. 1 • March 1996
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