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Minimax risk overl p -balls forl p -error
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  • Published: June 1994

Minimax risk overl p -balls forl p -error

  • David L. Donoho1 &
  • Iain M. Johnstone1 

Probability Theory and Related Fields volume 99, pages 277–303 (1994)Cite this article

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Summary

Consider estimating the mean vector θ from dataN n (θ,σ 2 I) withl q norm loss,q≧1, when θ is known to lie in ann-dimensionall p ball,p∈(0, ∞). For largen, the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp<q. Obvious exceptions aside, the limiting ratio equals 1 only ifp=q=2. Our arguments are mostly indirect, involving a reduction to a univariate Bayes minimax problem. Whenp<q, simple non-linear co-ordinatewise threshold rules are asymptotically minimax at small signal-to-noise ratios, and within a bounded factor of asymptotic minimaxity in general. We also give asymptotic evaluations of the minimax linear risk. Our results are basic to a theory of estimation in Besov spaces using wavelet bases (to appear elsewhere).

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Authors and Affiliations

  1. Department of Statistics, Stanford University, 94305, Stanford, CA, USA

    David L. Donoho & Iain M. Johnstone

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  1. David L. Donoho
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  2. Iain M. Johnstone
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Donoho, D.L., Johnstone, I.M. Minimax risk overl p -balls forl p -error. Probab. Th. Rel. Fields 99, 277–303 (1994). https://doi.org/10.1007/BF01199026

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  • Received: 23 June 1992

  • Revised: 13 November 1993

  • Issue Date: June 1994

  • DOI: https://doi.org/10.1007/BF01199026

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Mathematics Subject Classification (1985)

  • 62C20
  • 62F12
  • 62G20
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