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：我们提出了一种新的圆形和球形投影分布族——圆形投影柯西分布和球形投影柯西分布。我们证明了包裹柯西分布是圆形投影柯西分布的一种特殊情况。此外，我们提出了包裹柯西分布的一个泛化形式，这种形式包含一个额外的参数，可以改善分布的拟合性。在球形情况下，散布矩阵上的两个条件使分布具有椭圆对称性，这简化了其分析。投影分布具有良好的特性，如封闭形式的归一化常数和直接的随机值生成。可以通过最大似然估计其参数，并通过数值研究评估其偏差。利用真实数据集比较了提出的分布与现有模型，并表明提出的分布提供了更好的拟合。因此，圆形投影柯西分布和球形投影柯西分布是用于建模圆形和定向数据的有前途的选择。