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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).

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