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Uniform Behaviors of Random Polytopes under the Hausdorff Metric. (arXiv:1503.01504v2 [math.ST] UPDATED)
来源于:arXiv
We study the Hausdorff distance between a random polytope, defined as the
convex hull of i.i.d. random points, and the convex hull of the support of
their distribution. As particular examples, we consider uniform distributions
on convex bodies, densities that decay at a certain rate when approaching the
boundary of a convex body, projections of uniform distributions on higher
dimensional convex bodies and uniform distributions on the boundary of convex
bodies. We essentially distinguish two types of convex bodies: those with a
smooth boundary and polytopes. In the case of uniform distributions, we prove
that, in some sense, the random polytope achieves its best statistical accuracy
under the Hausdorff metric when the support has a smooth boundary and its worst
statistical accuracy when the support is a polytope. This is somewhat
surprising, since the exact opposite is true under the Nikodym metric. We prove
rate optimality of most our results in a minimax sense. In the case of uniform 查看全文>>