Cone distribution functions and quantiles for multivariate random variables

TL;DR

This paper introduces set-valued quantiles for multivariate distributions based on convex cones, extending univariate quantile properties and defining multivariate Value at Risk and stochastic orders.

Contribution

It proposes a novel set-valued quantile framework for multivariate data that leverages univariate distribution functions and explores their properties and applications.

Findings

Set-valued quantiles retain key properties of univariate quantiles.

Relationships established between multivariate quantiles, univariate quantiles, and depth functions.

Introduces multivariate Value at Risk and stochastic orders using the set-valued approach.

Abstract

Set-valued quantiles for multivariate distributions with respect to a general convex cone are introduced which are based on a family of (univariate) distribution functions rather than on the joint distribution function. It is shown that these quantiles enjoy basically all the properties of univariate quantile functions. Relationships to families of univariate quantile functions and to depth functions are discussed. Finally, a corresponding Value at Risk for multivariate random variables as well as stochastic orders are introduced via the set-valued approach.