Data Depth and Maximum Feasible Subsystems

Abstract

Various data depth measures were introduced in nonparametric statistics as multidimensional generalizations of ranks and of the median. A related problem in optimization is to find a maximum feasible subsystem, that is a solution satisfying as many constrainsts as possible, in a given system of linear inequalities. In this paper we give a unified framework for the main data depth measures such as the halfspace depth, the regression depth and the simplicial depth, and we survey the related results from nonparametric statistics, computational geometry, discrete geometry and linear optimization.