Probability Theory and Related Fields

, Volume 173, Issue 3–4, pp 1165–1196 | Cite as

Concentration of the empirical level sets of Tukey’s halfspace depth

  • Victor-Emmanuel BrunelEmail author


Tukey’s halfspace depth has attracted much interest in data analysis, because it is a natural way of measuring the notion of depth relative to a cloud of points or, more generally, to a probability measure. Given an i.i.d. sample, we investigate the concentration of upper level sets of the Tukey depth relative to that sample around their population version. We show that under some mild assumptions on the underlying probability measure, concentration occurs at a parametric rate and we deduce moment inequalities at that same rate. In a computational prospective, we study the concentration of a discretized version of the empirical upper level sets.


Tukey depth Level set Multivariate quantiles Support function Semi-infinite linear programming 

Mathematics Subject Classification



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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