Many datasets are observed on a finite set of equally spaced directions instead of the exact angles, such as the wind direction data. However, in the statistical literature, bivariate models are only available for continuous circular random variables. This article presents two bivariate circular distributions, namely bivariate wrapped geometric (BWG) and bivariate generalized wrapped geometric (BGWG), for analyzing bivariate discrete circular data. We consider wrapped geometric distributions and a trigonometric function to construct the models. The models are analytically tractable due to the exact closed-form expressions for the trigonometric moments. We thoroughly discuss the distributional properties of the models, including the interpretation of parameters and dependence structure. The estimation methodology based on maximizing the likelihood functions is illustrated for simulated datasets. Finally, the proposed distributions are utilized to analyze pairwise wind direction measurements obtained at different stations in India, and the interpretations for the fitted models are briefly discussed.

翻译：许多数据集是在一组等间距方向而非精确角度上观测得到的，例如风向数据。然而，在统计学文献中，二元模型仅适用于连续环形随机变量。本文提出了两种二元环形分布，即二元包裹几何分布（BWG）与二元广义包裹几何分布（BGWG），用于分析二元离散环形数据。我们利用包裹几何分布与三角函数构建了这些模型。由于三角矩具有精确的闭式表达式，这些模型在解析上是易于处理的。我们深入探讨了模型的分布性质，包括参数解释与相依结构。针对模拟数据集，阐述了基于似然函数最大化的估计方法。最后，利用所提出的分布分析了印度不同站点获取的成对风向测量数据，并对拟合模型的解释进行了简要讨论。