General notions of statistical depth function

Statistical depth Functions are being formulated ad hoc with increasing pop ularity in nonparametric inference for multivariate data. Here we introduce several general structures for depth functions, classify many existing exa mples as special cases, and establish results on the possession, or lack th ereof, of four key properties desirable for depth functions in general. Rou ghly speaking, these properties may be described as: affine invariance, max imality at center, monotonicity relative to deepest point, and vanishing at infinity. This provides a more systematic basis for selection of a depth f unction. In particular, from these and other considerations it is found tha t the halfspace depth behaves very well overall in comparison with various competitors.