Copulas from Order Statistics

TL;DR

This paper explores advanced properties of copulas derived from order statistics, introduces new copula classes, and evaluates their effectiveness in data fitting, highlighting the need for further development in multivariate copula models.

Contribution

It extends the class of copulas based on order statistics, introduces new copulas including one involving Bessel functions, and assesses their applicability in data fitting.

Findings

Some copulas can attain the Fréchet bound.

Existing multivariate copulas lack flexibility for general data fitting.

New copulas provide potential for better data modeling.

Abstract

A new class of copulas based on order statistics was introduced by Baker (2008). Here, further properties of the bivariate and multivariate copulas are described, such as that of likelihood ratio dominance (LRD), and further bivariate copulas are introduced that generalize the earlier work. One of the new copulas is an integral of a product of Bessel functions of imaginary argument, and can attain the Fr\'echet bound. The use of these copulas for fitting data is described, and illustrated with examples. It was found empirically that the multivariate copulas previously proposed are not flexible enough to be generally useful in data fitting, and further development is needed in this area.