Multivariate Order Statistics: Theory and Application

Abstract

This work revisits several proposals for the ordering of multivariate data via a prescribed depth function. We argue that one of these deserves special consideration, namely, Tukey's halfspace depth, which constructs nested convex sets via intersections of halfspaces. These sets provide a natural generalization of univariate order statistics to higher dimensions and exhibit consistency and asymptotic normality as estimators of corresponding population quantities. For absolutely continuous probability measures in , we present a connection between halfspace depth and the Radon transform of the density function, which is employed to formalize both the finite-sample and asymptotic probability distributions of the random nested sets. We review multivariate goodness-of-fit statistics based on halfspace depths, which were originally proposed in the projection pursuit literature. Finally, we demonstrate the utility of halfspace ordering as an exploratory tool by studying spatial data on maximum and minimum temperatures produced by a climate simulation model.