Donoho, David L.; Liu, Richard C.; MacGibbon, Brenda Minimax risk over hyperrectangles, and implications. (English) Zbl 0705.62018 Ann. Stat. 18, No. 3, 1416-1437 (1990). 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. Cited in 4 ReviewsCited in 75 Documents MSC: 62C20 Minimax procedures in statistical decision theory 62F10 Point estimation 62F12 Asymptotic properties of parametric estimators Keywords:estimating bounded normal mean; white noise; Ibragimov-Khas’minskij constant; Bernstein and Kolmogorov n-widths; standard Gaussian shift; orthosymmetric quadratically convex set; ellipsoids; hyperrectangles; minimax risk among linear estimates; minimax risk among truncated series estimates; Minimax linear estimators; nonlinear estimates; hardest rectangular subproblem; full problem PDFBibTeX XMLCite \textit{D. L. Donoho} et al., Ann. Stat. 18, No. 3, 1416--1437 (1990; Zbl 0705.62018) Full Text: DOI