## 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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### References:

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