Minimax Risk Over Hyperrectangles, and Implications

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

Ann. Statist., Tome 18 (1990) no. 1, p. 1416-1437 / Harvested from Project Euclid

Résumé

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.

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Publié le : 1990-09-14
Classification:  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,  62C20,  62F10,  62F12