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.

翻译：连接函数作为表示多变量间依赖结构的统计模型已在众多应用中广泛流行。在预先给定有限约束条件下，最小信息连接函数是在库尔巴克-莱布勒散度度量下最接近均匀连接函数的连接函数。前人研究主要关注斯皮尔曼秩相关系数等矩的期望作为约束条件，此类连接函数可通过凸规划求得最优解。然而，其他类型相关系数在该框架下尚未被研究。本文提出MICK——一种以肯德尔秩相关系数为约束的新型最小信息连接函数。尽管该连接函数定义为非凸优化问题的解，我们证明当相关系数足够小时，该连接函数的唯一性得以保证。我们还证明棋盘连接函数族可表示为非正交向量空间。通过分析MICK的局部与全局依赖特征，本研究实现了最小信息连接函数相关成果的统一。