Halfspace Depth and Regression Depth Characterize the Empirical Distribution

For multivariate data, the halfspace depth function can be seen as a natural and affine equivariant generalization of the univariate empirical cdf. For any multivariate data set, we show that the resulting halfspace depth function completely determines the empirical distribution. We do this by actua...

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Veröffentlicht in:Journal of multivariate analysis 1999-04, Vol.69 (1), p.135-153
Hauptverfasser: Struyf, Anja J, Rousseeuw, Peter J
Format: Artikel
Sprache:eng
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Zusammenfassung:For multivariate data, the halfspace depth function can be seen as a natural and affine equivariant generalization of the univariate empirical cdf. For any multivariate data set, we show that the resulting halfspace depth function completely determines the empirical distribution. We do this by actually reconstructing the data points from the depth contours. The data need not be in general position. Moreover, we prove the same property for regression depth.
ISSN:0047-259X
1095-7243
DOI:10.1006/jmva.1998.1804