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October 2003 A scatter matrix estimate based on the zonotope
Gleb A. Koshevoy, Jyrki Möttönen, Hannu Oja
Ann. Statist. 31(5): 1439-1459 (October 2003). DOI: 10.1214/aos/1065705114


We introduce a new scatter matrix functional which is a multivariate affine equivariant extension of the mean deviation $E(|x-\mbox{Med}(x)|)$. The estimate is constructed using the data vectors (centered with the multivariate Oja median) and their angular distances. The angular distance is based on Randles interdirections. The new estimate is called the zonoid covariance matrix (the ZCM), as it is the regular covariance matrix of the centers of the facets of the zonotope based on the data set. There is a kind of symmetry between the zonoid covariance matrix and the affine equivariant sign covariance matrix; interchanging the roles of data vectors and hyperplanes yields the sign covariance matrix as the zonoid covariance matrix. It turns out that the symmetry relies on the zonoid of the distribution and its projection body which is also a zonoid.) The influence function and limiting distribution of the new scatter estimate, the ZCM, are derived to consider the robustness and efficiency properties of the estimate. Finite-sample efficiencies are studied in a small simulation study. The influence function of the ZCM is unbounded (linear in the radius of the contamination vector) but less influential in the tails than that of the regular covariance matrix (quadratic in the radius). The estimate is highly efficient in the multivariate normal case and performs better than the regular covariance matrix for heavy-tailed distributions.


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Gleb A. Koshevoy. Jyrki Möttönen. Hannu Oja. "A scatter matrix estimate based on the zonotope." Ann. Statist. 31 (5) 1439 - 1459, October 2003.


Published: October 2003
First available in Project Euclid: 9 October 2003

zbMATH: 1046.62058
MathSciNet: MR2012821
Digital Object Identifier: 10.1214/aos/1065705114

Primary: 62H12

Keywords: Interdirection , multivariate mean deviation , multivariate median , multivariate signs , parallelotope , scatter matrix , sign covariance matrix , zonoid , zonoid covariance matrix , zonotope

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 5 • October 2003
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