A0309
Title: Distributionally robust halfspace depth
Authors: Jevgenijs Ivanovs - University of Aarhus (Denmark)
Pavlo Mozharovskyi - LTCI, Telecom Paris, Institut Polytechnique de Paris (France) [presenting]
Abstract: Statistical data depth function measures the centrality of an observation with respect to a distribution or a data set by a number between 0 and 1 while satisfying certain postulates regarding invariance, monotonicity, and convexity. It constitutes a contemporary domain of rapid development to meet growing demand in various areas of industry, economy, social sciences, etc. Being one of the most studied depth notions, Tukey's halfspace depth can be seen as a stochastic program, and as such, it suffers from the optimizer's curse, so that a limited training sample may easily result in a poor out-of-sample performance. We propose a generalized halfspace depth concept relying on the recent advances in distributionally robust optimization, where every halfspace is examined using the respective worst-case distribution in the Wasserstein ball centered at the empirical law. This new depth can be seen as a smoothed and regularized classical halfspace depth, which is retrieved as the radius of the Wasserstein ball vanishes. It inherits the main properties of the latter and, additionally, enjoys various new attractive features such as continuity and strict positivity beyond the convex hull of the support. We provide numerical illustrations of the new depth and its advantages and develop some fundamental theory. In particular, we study the upper-level sets and the median region, including their breakdown properties.
