Copulas with maximum entropy

We shall find a multi-dimensional checkerboard copula of maximum entropy that matches an observed set of grade correlation coefficients. This problem is formulated as the maximization of a concave function on a convex polytope. Under mild constraint qualifications we show that a unique solution exists in the core of the feasible region. The theory of Fenchel duality is used to reformulate the problem as an unconstrained minimization which is well solved numerically using a Newton iteration. Finally, we discuss the numerical calculations for some hypothetical examples and describe how this work can be applied to the modelling and simulation of monthly rainfall.