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2014 Data-based decision rules about the convexity of the support of a distribution
Pedro Delicado, Adolfo Hernández, Gábor Lugosi
Electron. J. Statist. 8(1): 96-129 (2014). DOI: 10.1214/14-EJS877


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.


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Pedro Delicado. Adolfo Hernández. Gábor Lugosi. "Data-based decision rules about the convexity of the support of a distribution." Electron. J. Statist. 8 (1) 96 - 129, 2014.


Published: 2014
First available in Project Euclid: 10 February 2014

zbMATH: 1282.62022
MathSciNet: MR3165435
Digital Object Identifier: 10.1214/14-EJS877

Primary: 62G10
Secondary: 62H30

Keywords: bootstrap subsampling , dimensionality reduction , Discernibility between hypotheses , ISOMAP , set estimation , U-statistics

Rights: Copyright © 2014 The Institute of Mathematical Statistics and the Bernoulli Society


Vol.8 • No. 1 • 2014
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