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Minimax risk over \(l_ p\)-balls for \(l_ q\)-error. (English) Zbl 0802.62006
Summary: Consider estimating the mean vector \(\theta\) from data \(N_ n (\theta, \sigma^ 2 I)\) with \(l_ q\) norm loss, \(q\geq 1\), when \(\theta\) is known to lie in an \(n\)-dimensional \(l_ p\) ball, \(p\in (0,\infty)\). 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 nonlinear coordinatewise 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).

MSC:
62C20 Minimax procedures in statistical decision theory
62F12 Asymptotic properties of parametric estimators
62G20 Asymptotic properties of nonparametric inference
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