Abstract
Given $n$ independent, identically distributed random vectors in $\mathbb{R}^{d}$, drawn from a common density $f$, one wishes to find out whether the support of $f$ is convex or not. In this paper we describe a decision rule which decides correctly for sufficiently large $n$, with probability $1$, whenever $f$ is bounded away from zero in its compact support. We also show that the assumption of boundedness is necessary. The rule is based on a statistic that is a second-order $U$-statistic with a random kernel. Moreover, we suggest a way of approximating the distribution of the statistic under the hypothesis of convexity of the support. The performance of the proposed method is illustrated on simulated data sets. As an example of its potential statistical implications, the decision rule is used to automatically choose the tuning parameter of ISOMAP, a nonlinear dimensionality reduction method.
Highlights
The main objective of this paper is to investigate the possibility of constructing consistent decision rules for the convexity of the support
We show that consistent decision rules exist whenever f is bounded away from zero on its support and some other mild regularity conditions are satisfied
It is shown that it is impossible to design a decision rule that behaves asymptotically correctly for all bounded densities of bounded support. This shows that an assumption like the density being bounded away from zero on its support is necessary for consistent decision rules
Summary
Let X1, . . . , Xn be i.i.d. vectors drawn from the probability distribution μ on Rd. If the support is not convex we expect to have a large number of pairs (Xi, Xj) such that the closest point to (Xi + Xj)/2 is far away. Every term in the second sum on the right-hand is in [−1, 0] and is not zero only if h(1)(i, j) = k This implies that (Un. denoting by Nk = 1⁄2 (i,j):i
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