We propose a new family of projected distributions on the circle and the sphere, the circular and the spherical projected Cauchy distributions. We show that the wrapped Cauchy distribution is a special of the circular projected Cauchy distribution. Further, a generalization of the wrapped Cauchy distribution is proposed, which includes an extra parameter that improves the fit of the distribution. For the spherical case, the imposition of two conditions on the scatter matrix makes the distribution elliptically symmetric, which simplifies its analysis. The projected distributions have nice features, such as closed-form normalizing constant and straightforward random value generation. The parameters of the distributions can be estimated via maximum likelihood, and their bias will be assessed through numerical studies. The proposed distributions have been compared to existing models using real data sets, and are shown to provide a better fit. Therefore, the circular projected and spherical projected Cauchy distributions are promising alternatives for modeling circular and directional data.

翻译：我们提出了一类新的圆和球面上的投影分布族，即圆投影柯西分布和球面投影柯西分布。研究表明，缠绕柯西分布是圆投影柯西分布的一个特例。进一步地，我们提出了一种缠绕柯西分布的推广形式，该推广引入了一个额外参数以改善分布的拟合效果。对于球面情形，在散射矩阵上施加两个条件可使分布具有椭圆对称性，从而简化分析。该投影分布具有优良特性，例如归一化常数具有闭合形式，且随机值生成过程直接明了。分布参数可通过极大似然估计法进行估计，并通过数值研究评估其偏差。利用真实数据集将所提分布与现有模型进行了比较，结果表明所提分布具有更好的拟合效果。因此，圆投影柯西分布和球面投影柯西分布是环形与方向数据建模中极具前景的替代方案。