Maximum entropy distribution of order statistics with given marginals

TL;DR

This paper characterizes the maximum entropy distribution of ordered random vectors with specified marginals, providing explicit formulas and conditions for existence and uniqueness.

Contribution

It offers a necessary and sufficient condition for such distributions to exist and derives the explicit density of the maximum entropy distribution with given marginals.

Findings

Provides explicit density formula for maximum entropy distribution.

Establishes existence and uniqueness conditions.

Connects copulas with order statistics and given marginals.

Abstract

We consider distributions of ordered random vectors with given one-dimensional marginal distributions. We give an elementary necessary and sufficient condition for the existence of such a distribution with finite entropy. In this case, we give explicitly the density of the unique distribution which achieves the maximal entropy and compute the value of its entropy. This density is the unique one which has a product form on its support and the given one-dimensional marginals. The proof relies on the study of copulas with given one-dimensional marginal distributions for its order statistics.