Lift-zonoid and Multivariate Depths

Summary

Tukey (1975) proposed the halfspace depth concept as a geometrical tool to handle measures. However, only recently (Koshevoy, 1999b; Struyf and Rousseeuw, 1999), it was shown that, for the class of atomic measures, this depth determines the measure. Here we extend this characterization result for the class of absolutely continuous measures for which the function exp(<p, x>) is integrable with any $p \in {{\Bbb R}^d}$. Three issues play a key role in proving this characterization. The first, the Tukey median has a depth $\geqslant 1/(K + 1)$ for any k-variate distribution. The second, let two measures μ and v have the same Tukey depth. Then the restrictions of these measures to any trimmed region are measures with identical Tukey depths. The third, a relation between Tukey depth of a measure with compact support and some projections of lift-zonoid. This relation allows to use the support theorem for the Radon transform.

We also show that, for the class of measures with full-dimensional convex hull of the support, the Oja depth determines the measure. The proof of this results are based on some relation between the Oja depth and projections of the lift zonoid. This relation allows us to use another result of integral geometry: the uniqueness theorem of Alexandrov (1937).