We introduce a novel family of projected distributions on the circle and the sphere, namely the circular and spherical projected Cauchy distributions, as promising alternatives for modelling circular and spherical data. The circular distribution encompasses the wrapped Cauchy distribution as a special case, while featuring a more convenient parameterisation. We also propose a generalised wrapped Cauchy distribution that includes an extra parameter, enhancing the fit of the distribution. In the spherical context, we impose two conditions on the scatter matrix of the Cauchy distribution, resulting in an elliptically symmetric distribution. Our projected distributions exhibit attractive properties, such as a closed-form normalising constant and straightforward random value generation. The distribution parameters can be estimated using maximum likelihood, and we assess their bias through numerical studies. Further, we compare our proposed distributions to existing models with real datasets, demonstrating equal or superior fitting both with and without covariates.

翻译：本文提出了一类新颖的圆形与球形投影分布族——圆形与球形投影柯西分布，作为建模圆形与球形数据的有前景的替代方案。该圆形分布将缠绕柯西分布作为特例包含在内，同时具备更便捷的参数化形式。我们还提出了一种广义缠绕柯西分布，通过引入额外参数增强了分布的拟合能力。在球形场景中，我们对柯西分布的散度矩阵施加两个约束条件，从而得到一种椭圆对称分布。我们提出的投影分布展现出诸多优良特性，例如具有闭合形式的归一化常数以及简便的随机值生成方法。分布参数可通过最大似然法进行估计，并通过数值研究评估了其偏差。此外，我们使用真实数据集将所提出的分布与现有模型进行比较，结果表明无论是否包含协变量，其拟合效果均达到或优于现有模型。