MINIMAX RISK OF L(P)-BALLS FOR L(Q)-ERROR

Consider estimating the mean vector theta from data N(n)(theta, sigma2I) with l(q) norm loss, q greater-than-or-equal-to 1, when theta is known to lie in an n-dimensional l(p) ball, p is-an-element-of (0, infinity). 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).