Finite Representation of Quantile Sets for Multivariate Data via Vector Linear Programming

TL;DR

This paper introduces a method to compute multivariate empirical quantiles and Tukey depth regions using vector linear programming, extending univariate techniques to higher dimensions.

Contribution

It presents a novel approach that uses vector linear programming to efficiently compute multivariate quantiles and depth regions, bridging a gap in multivariate statistical analysis.

Findings

Multivariate quantiles can be obtained via vector linear programming.

The method simplifies computation of Tukey depth regions.

Provides a new computational framework for cone quantile sets.

Abstract

Empirical quantiles for finitely distributed univariate random variables can be obtained by solving a certain linear program. It is shown in this short note that multivariate empirical quantiles can be obtained in a very similar way by solving a vector linear program. This connection provides a new approach for computing Tukey depth regions and more general cone quantile sets.