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On resampling schemes for polytopes

Part of: Nonparametric inference

Published online by Cambridge University Press:  11 December 2019

Weinan Qi  and
Mahmoud Zarepour
Weinan Qi*
Affiliation:
University of Ottawa
Mahmoud Zarepour*
Affiliation:
University of Ottawa
*
* Postal address: Department of Mathematics and Statistics, University of Ottawa, Canada.
* Postal address: Department of Mathematics and Statistics, University of Ottawa, Canada.
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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. To approximate the distribution of the functionals of convex hulls, asymptotic theory plays a crucial role. Unfortunately most of the asymptotic results are computationally intractable. To address this computational intractability, we consider consistent bootstrapping schemes for certain cases. Let $S_n=\{X_i\}_{i=1}^{n}$ be a sequence of independent and identically distributed random points uniformly distributed on an unknown convex set in $\mathbb{R}^{d}$ ($d\ge 2$ ). We suggest a bootstrapping scheme that relies on resampling uniformly from the convex hull of $S_n$ . Moreover, the resampling asymptotic consistency of certain functionals of convex hulls is derived under this bootstrapping scheme. In particular, we apply our bootstrapping technique to the Hausdorff distance between the actual convex set and its estimator. For $d=2$ , we investigate the asymptotic consistency of the suggested bootstrapping scheme for the area of the symmetric difference and the perimeter difference between the actual convex set and its estimate. In all cases the consistency allows us to rely on the suggested resampling scheme to study the actual distributions, which are not computationally tractable.

Keywords

Convex hullrandom polytopesbootstrapasymptotic

MSC classification

Primary: 62G09: Resampling methods
Secondary: 62G20: Asymptotic properties 62G32: Statistics of extreme values; tail inference
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 4 , December 2019 , pp. 959 - 980
Copyright
© Applied Probability Trust 2019 

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