Concentration of the empirical level sets of Tukey’s halfspace depth
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Author(s)
Brunel, Victor-Emmanuel
Date Issued
May 11, 2018
Publisher
Springer Berlin Heidelberg
Version
Author's final manuscript
Abstract
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00440-018-0850-0
