Minimax risk overl
p
-balls forl
p
-error |
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Authors: | David L Donoho Iain M Johnstone |
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Institution: | (1) Department of Statistics, Stanford University, 94305 Stanford, CA, USA |
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Abstract: | Summary Consider estimating the mean vector from dataN
n
( ,
2
I) withl
q
norm loss,q 1, when 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. 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, 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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Keywords: | 62C20 62F12 62G20 |
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