On Resampling Schemes for Uniform Polytopes

Abstract

The convex hull of a sample is used to approximate the support of the underlying distribution. This approximation has many practical implications in real life. For example, approximating the boundary of a finite set is used by many authors in environmental studies and medical research. To approximate the functionals of convex hulls, asymptotic theory plays a crucial role. Unfortunately, the asymptotic results are mostly very complicated. To address this complication, we suggest a consistent bootstrapping scheme for certain cases. Our resampling technique is used for both semi-parametric and non-parametric cases. Let X1,X2,...,Xn be a sequence of i.i.d. random points uniformly distributed on an unknown convex set. Our bootstrapping scheme relies on resampling uniformly from the convex hull of X1,X2,...,Xn. In this thesis, we study the asymptotic consistency of certain functionals of convex hulls. In particular, we apply our bootstrapping technique to the Hausdorff distance between the actual convex set and its estimator. We also provide a conjecture for the application of our bootstrapping scheme to Gaussian polytopes. Moreover, some other relevant consistency results for the regular bootstrap are developed.