Frank copula is minimum information copula under fixed Kendall’s <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg" display="inline" id="d1e22"><mml:mi>τ</mml:mi></mml:math>

In this work, we demonstrate that the Frank copula is the minimum information copula under fixed Kendall’s τ (MICK), both theoretically and numerically. First, we explain that both MICK and the Frank density follow the hyperbolic Liouville equation. Subsequently, 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 τ as the sole available information about the true distribution, based on the entropy maximization principle. • The minimum information copula under fixed Kendall’s τ is identical to the Frank copula. • Both of these two copulas follow the well-known hyperbolic Liouville equation. • This result can be confirmed both theoretically and numerically.