Abstract

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.

Keywords

MinimaxMathematicsQuadratic growthEstimatorMinimax estimatorMinimax approximation algorithmGaussianApplied mathematicsHeuristicRegular polygonCombinatoricsStatisticsMathematical optimizationMathematical analysisMinimum-variance unbiased estimatorGeometry

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Publication Info

Year
1990
Type
article
Volume
18
Issue
3
Citations
193
Access
Closed

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David L. Donoho, Richard C. Liu, Brenda MacGibbon (1990). Minimax Risk Over Hyperrectangles, and Implications. The Annals of Statistics , 18 (3) . https://doi.org/10.1214/aos/1176347758

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DOI
10.1214/aos/1176347758