Abstract:
Consider the problem of estimating a closed convex set $G$ on the plane, given a sample from the uniform distribution on this set. We assume that $G$ belongs either to the class of all closed convex subsets of a fixed circle with Lebesgue measure separated from 0, or to a smaller class of all convex sets with smooth boundaries having curvature radius $\geq R_0$. We study the asymptotics of the minimax risk over these classes, assuming that the distance between the estimator and the true set is measured in the Hausdorff metric. We show that the asymptotics of the risks are different for these classes, namely, $O((n/(\log n))^{-1/2})$ and $O((n/(\log n))^{-2/3})$, respectively. Moreover, for the class of smooth convex sets $G$ we obtain a sharp evaluation of the minimax risk, with the ratio of lower and upper bounds $\approx 0,96$.
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