Copulas for maxmin systems

TL;DR

This paper derives explicit formulas for copulas of systems formed by maxima and minima of subvectors of a multivariate distribution, with applications to shock models and order statistics.

Contribution

It provides closed-form expressions for maxmin copulas under mild conditions, including the i.i.d. case where they become universal and condition-free.

Findings

Explicit copula formulas for maxmin systems.

Universal maxmin copulas in the i.i.d. case.

Applications to shock models and order statistics.

Abstract

Under a mild condition we give closed-form expressions for copulas of systems that consist of maxima and of minima of subvectors of a given random vector $X$ with continuous marginals. Said expressions appear explicit in the copula of $X$ and the mentioned condition is for example met when the law of $X$ admits a strictly positive density with respect to Lebesgue measure. In the i.i.d. case these "maxmin" copulae become universal and the conditions on their validity can be dropped entirely. Our main motivation comes from applications to shock models that arise in multivariate survival theory. Another application is to order statistics copulas.