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 modeling circular and directional data. The circular distribution encompasses the wrapped Cauchy distribution as a special case, featuring a more convenient parameterisation. Next, we 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, resulting in an elliptically symmetric distribution. Our projected distributions exhibit attractive properties, such as closed-form normalising constants and straightforward random value generation. The distribution parameters can be estimated using maximum likelihood and we assess their bias through numerical studies. We compare our proposed distributions to existing models with real data sets, demonstrating superior fit both with and without covariates.

翻译：我们提出了一类新颖的圆环与球面投影分布族，即圆形和球面投影柯西分布，作为圆形与方向数据建模的有力替代方案。其中，圆形分布将包裹柯西分布作为特例，并具有更便捷的参数化形式。随后，我们提出了一种包含额外参数的广义包裹柯西分布，以提升分布的拟合性能。在球面情境下，我们对散射矩阵施加两个约束条件，从而得到椭圆对称分布。所提出的投影分布具备吸引人的特性，如归一化常数的闭合形式表达及随机值生成的简便性。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其偏差。我们利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，新分布均展现更优的拟合效果。