Copulas have gained widespread popularity as statistical models to represent dependence structures between multiple variables in various applications. The minimum information copula, given a finite number of constraints in advance, emerges as the copula closest to the uniform copula when measured in Kullback-Leibler divergence. In prior research, the focus has predominantly been on constraints related to expectations on moments, including Spearman's $\rho$. This approach allows for obtaining the copula through convex programming. However, the existing framework for minimum information copulas does not encompass non-linear constraints such as Kendall's $\tau$. To address this limitation, we introduce MICK, a novel minimum information copula under fixed Kendall's $\tau$. We first characterize MICK by its local dependence property. Despite being defined as the solution to a non-convex optimization problem, we demonstrate that the uniqueness of this copula is guaranteed when the correlation is sufficiently small. Additionally, we provide numerical insights into applying MICK to real financial data.

翻译：联结函数作为统计模型在多种应用中广泛用于表示多变量间的依赖结构。最小信息联结函数在给定有限约束条件下，以Kullback-Leibler散度度量时，被定义为最接近均匀联结函数的联结函数。以往研究主要聚焦于矩期望约束（包括Spearman秩相关系数ρ），此类约束可通过凸规划获得联结函数。然而，现有最小信息联结函数框架无法涵盖Kendall秩相关系数τ等非线性约束。针对这一局限，我们提出MICK——一种在固定Kendall秩相关系数τ下的新型最小信息联结函数。首先通过局部依赖性质刻画MICK的特征。尽管该联结函数被定义为非凸优化问题的解，我们证明当相关性足够小时其唯一性得以保证。此外，我们提供将MICK应用于真实金融数据的数值分析。