Minimax risk over / p-balls for / q-error

Consider estimating the mean vector 0 from data N,(O, a2I) with l~ norm loss, q > 1, when 0 is known to lie in an n-dimensional 1 v ball, p e (0, oe ). For large n, the ratio of minimax linear risk to minimax risk can be arbitrarily large if p < q. Obvious exceptions aside, the limiting ratio equals 1 only if p = q = 2. Our arguments are mostly indirect, involving a reduction to a univariate Bayes minimax problem. When p < 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).