Some questions in mathematics are not answered for quite some time, but just sidestepped. One of those questions is the following: What is the quantile of a multi-dimensional random variable? The "sidestepping" in this case produced so-called depth functions and depth regions, and the most prominent among them is the halfspace depth invented by Tukey in 1975, a very popular tool in statistics. When it comes to the definition of multivariate quantiles, depth functions replace cummulative distribution functions, and depth regions provide potential candidates for quantile vectors. However, Tukey depth functions, for example, do not share all features with (univariate) cdf's and do not even generalize them.
On the other hand, the naive definition of quantiles via the joint distribution function turned out to be not very helpful for statistical purposes, although it is still in use to define multivariate V@Rs (Embrechts and others) as well as stochastic dominance orders (Muller/Stoyan and others).
The crucial point and an obstacle for substantial progress for a long time is the missing (total) order for the values of a multi-dimensional random variable. On the other hand, (non-total) orders appear quite natural in financial models with proportional transaction costs (a.k.a. the Kabanov market) in form of solvency cones.
We propose new concepts for multivariate ranking functions with features very close to univariate cdf's and for set-valued quantile functions which, at the same time, generalize univariate quantiles as well as Tukey's halfspace depth regions. Our constructions are designed to deal with general vector orders for the values of random variables, and they produce unambigious lower and upper multivariate quantiles, multivariate V@Rs as well as a multivariate first order stochastic dominance relation. Financial applications to markets with frictions are discussed as well as many other examples and pictures which show the interesting geometric features of the new quantile sets.
The talk is based on: AH Hamel, D Kostner, Cone distribution functions and quantiles for multivariate random variables , J. Multivariate Analysis 167, 2018[-]