Frank copula is minimum information copula under fixed Kendall's $\tau$

TL;DR

This paper proves that the Frank copula is the minimum information copula for a fixed Kendall's tau, establishing its optimality in dependence modeling through theoretical and numerical analysis.

Contribution

It demonstrates that the Frank copula uniquely satisfies the hyperbolic Liouville equation and is the minimum information copula under fixed Kendall's tau, linking entropy maximization to its selection.

Findings

Frank copula satisfies the hyperbolic Liouville equation.

Frank copula is the unique solution to the Liouville equation.

Selecting the Frank copula aligns with entropy maximization under fixed Kendall's tau.

Abstract

In dependence modeling, various copulas have been utilized. Among them, the Frank copula has been one of the most typical choices due to its simplicity. In this work, we demonstrate that the Frank copula is the minimum information copula under fixed Kendall's $\tau$ (MICK), both theoretically and numerically. First, we explain that both MICK and the Frank density follow the hyperbolic Liouville equation. Moreover, we show that the copula density satisfying the Liouville equation is uniquely the Frank copula. Our result asserts that selecting the Frank copula as an appropriate copula model is equivalent to using Kendall's $\tau$ as the sole available information about the true distribution, based on the entropy maximization principle.