The depth function of a population distribution

Tukey (1975) introduced the notion of halfspace depth in a data analytic context, as a multivariate analog of rank relative to a finite data set. Here we focus on the depth function of an arbitrary probability distribution on Â^p, and even of a non-probability measure. The halfspace depth of any point / in Â^p is the smallest measure of a closed halfspace that contains /. We review the properties of halfspace depth, enriched with some new results. For various measures, uniform as well as non-uniform, we derive an expression for the depth function. We also compute the Tukey median, which is the / in which the depth function attains its maximal value. Various interesting phenomena occur. For the uniform distribution on a triangle, a square or any regular polygon, the depth function has ridges that correspond to an 'inversion' of depth contours. And for a product of Cauchy distributions, the depth contours are squares. We also consider an application of the depth function to voting theory.