Minimax Risk Over Hyperrectangles, and Implications

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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APA Style

                            
                                    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

Identifiers

DOI: 10.1214/aos/1176347758