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.

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