# Bivariate distributions with ordered marginals

## TL;DR

This paper characterizes dependency structures between two stochastically ordered variables using copulas, providing extremal measures, explicit joint distributions, and extensions to multivariate and partially ordered spaces.

## Contribution

It offers a comprehensive characterization of copulas compatible with stochastic order, including extremal measures and maximum entropy distributions, with multivariate extensions.

## Key findings

- Closed-form extremal Kendall's tau and Spearman's rho
- Explicit maximum entropy joint distribution
- Extensions to multivariate and partially ordered spaces

## Abstract

This paper provides a characterization of all possible dependency structures between two stochastically ordered random variables. The answer is given in terms of copulas that are compatible with the stochastic order and the marginal distributions. The extremal values for Kendall's $\tau$ and Spearman's $\rho$ for all these copulas are given in closed form. We also find an explicit form for the joint distribution with the maximal entropy. A multivariate extension and a generalization to random elements in partially ordered spaces are also provided.