On the minimum information checkerboard copulas under fixed Kendall's rank correlation

Copulas have become very popular as a statistical model to represent dependence structures between multiple variables in many applications. Given a finite number of constraints in advance, the minimum information copula is the closest to the uniform copula when measured in Kullback-Leibler divergence. For these constraints, the expectation of moments such as Spearman's rho are mostly considered in previous researches. These copulas are obtained as the optimal solution to convex programming. On the other hand, other types of correlation have not been studied previously in this context. In this paper, we present MICK, a novel minimum information copula where Kendall's rank correlation is specified. Although this copula is defined as the solution to non-convex optimization problem, we show that the uniqueness of this copula is guaranteed when correlation is small enough. We also show that the family of checkerboard copulas admits representation as non-orthogonal vector space. In doing so, we observe local and global dependencies of MICK, thereby unifying results on minimum information copulas.