We propose the extension of Fr\'{e}chet-Hoeffding copula bounds for circular data. The copula is a powerful tool for describing the dependency of random variables. In two dimensions, the Fr\'{e}chet-Hoeffding upper (lower) bound indicates the perfect positive (negative) dependence between two random variables. However, for circular random variables, the usual concept of dependency is not accepted because of their periodicity. In this work, we redefine Fr\'{e}chet-Hoeffding bounds and consider modified Fr\'{e}chet and Mardia families of copulas for modelling the dependency of two circular random variables. Simulation studies are also given to demonstrate the behavior of the model.
翻译：我们提议延长循环数据的 Fr\'{e}chet-Hoffing cogula 边框。 copula 是描述随机变量依赖性的有力工具。 在两个方面, Fr\'{e}chet-Hoffding 上( 下) 边框显示两个随机变量之间的完全正( 负) 依赖性。 但是,对于循环随机变量,通常的依赖性概念因其周期性而不被接受。 在这项工作中, 我们重新定义 Fr\' {e}chet- hoffing 边框, 并考虑修改 Fr\'{e}chet 和 Mardia 边框组来模拟两个循环随机变量的依赖性。 还进行了模拟研究, 以演示模型的行为 。
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