, Volume 99, Issue 2, pp 277-303

Minimax risk overl p -balls forl p -error

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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. 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).