Depth in Infinite-dimensional Spaces

Abstract

Depth is a statistical tool that aims to introduce sensible data-dependent ordering of points in multivariate / function spaces. In theory, this should allow construction of statistical procedures based on ranks, orderings, or quantiles for multi-dimensional data. Some of the natural properties a depth should satisfy in finite-dimensional spaces however lose tractability and appeal as the dimension grows. We introduce the depth in finite-dimensional spaces, and outline particular difficulties one faces when attempting to generalize depths to the situation of functional, or other infinite-dimensional data.