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Minimax risk over hyperrectangles, and implications. (English) Zbl 0705.62018

Summary: Consider estimating the mean of a standard Gaussian shift when that mean is known to lie in an orthosymmetric quadratically convex set in \(l_ 2\). Such sets include ellipsoids, hyperrectangles and \(l_ p\)-bodies with \(p>2\). The minimax risk among linear estimates is within 25 % of the minimax risk among all estimates. The minimax risk among truncated series estimates is within a factor 4.44 of the minimax risk. This implies that the difficulty of estimation - a statistical quantity - is measured fairly precisely by the n-widths - a geometric quantity.
If the set is not quadratically convex, as in the case of \(l_ p\)-bodies with \(p<2\), things change appreciably. Minimax linear estimators may be outperformed arbitrarily by nonlinear estimates. The (ordinary, Kolmogorov) n-widths still determine the difficulty of linear estimation, but the difficulty of nonlinear estimation is tied to the (inner, Bernstein) n-widths, which can be far smaller.
Essential use is made of a new heuristic: that the difficulty of the hardest rectangular subproblem is equal to the difficulty of the full problem.

MSC:

62C20 Minimax procedures in statistical decision theory
62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
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