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Halfspace Depth and Regression Depth Characterize the Empirical Distribution

Anja J. Struyf and Peter Rousseeuw ()

Journal of Multivariate Analysis, 1999, vol. 69, issue 1, 135-153

Abstract: 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.

Keywords: location depth; multivariate ranking; reconstruction algorithm; regression depth (search for similar items in EconPapers)
Date: 1999
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Handle: RePEc:eee:jmvana:v:69:y:1999:i:1:p:135-153