Statistics Seminar: STA 290
Thursday, May 9th, 2013 at 4:10pm, MSB 1147 (Colloquium Room)
Refreshments at 3:30pm in MSB 4110 (Statistics Lounge)
Speaker: Pedro Delicado, Universitat Politècnica de Catalunya, Barcelona (Spain)
Title: "Data-based decision rules about the convexity of the support of a distribution"
Abstract: Given n independent, identically distributed random vectors in 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. (Joint work with Gabor Lugosi and Adolfo Hernandez)