A scatter matrix estimate based on the zonotope.

*(English)*Zbl 1046.62058Summary: We introduce a new scatter matrix functional which is a multivariate affine equivariant extension of the mean deviation \(E(| x-\text{Med}(x) |)\). The estimate is constructed using the data vectors (centered with the multivariate Oja median) and their angular distances. The angular distance is based on the Randles interdirections [R. H. Randles, J. Am. Stat. Assoc. 84, No. 408, 1045–1050 (1989; Zbl 0702.62039)]. 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.

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.

##### MSC:

62H12 | Estimation in multivariate analysis |

62H05 | Characterization and structure theory for multivariate probability distributions; copulas |

##### Keywords:

interdirection; parallelotope; multivariate mean deviation; multivariate median; multivariate signs; sign covariance matrix; scatter matrix; zonoid; zonoid covariance matrix; zonotope
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\textit{G. A. Koshevoy} et al., Ann. Stat. 31, No. 5, 1439--1459 (2003; Zbl 1046.62058)

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##### References:

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