The Annals of Statistics

Minimax Risk Over Hyperrectangles, and Implications

David L. Donoho, Richard C. Liu, and Brenda MacGibbon

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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 out-performed 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.

Article information

Ann. Statist., Volume 18, Number 3 (1990), 1416-1437.

First available in Project Euclid: 12 April 2007

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Primary: 62C20: Minimax procedures
Secondary: 62F10: Point estimation 62F12: Asymptotic properties of estimators

Estimating a bounded normal mean estimating a function observed with white noise hardest rectangular subproblems Ibragimov-Has'minskii constant quadratically convex sets Bernstein and Kolmogorov $n$-widths


Donoho, David L.; Liu, Richard C.; MacGibbon, Brenda. Minimax Risk Over Hyperrectangles, and Implications. Ann. Statist. 18 (1990), no. 3, 1416--1437. doi:10.1214/aos/1176347758.

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