In 2020, two novel distributions for the analysis of directional data were introduced: the spherical Cauchy distribution and the Poisson kernel-based distribution. This paper provides a detailed exploration of both distributions within various analytical frameworks. To enhance the practical utility of these distributions, alternative parametrizations that offer advantages in numerical stability and parameter estimation are presented, such as implementation of the Newton-Raphson algorithm for parameter estimation, while facilitating a more efficient and simplified approach in the regression framework. Additionally, a two-sample location test based on the log-likelihood ratio test is introduced. This test is designed to assess whether the location parameters of two populations can be assumed equal. The maximum likelihood discriminant analysis framework is developed for classification purposes, and finally, the problem of clustering directional data is addressed, by fitting finite mixtures of Spherical Cauchy or Poisson kernel-based distributions. Empirical validation is conducted through comprehensive simulation studies and real data applications, wherein the performance of the spherical Cauchy and Poisson kernel-based distributions is systematically compared.

翻译：2020年，两种用于方向数据分析的新分布被提出：球面柯西分布与基于泊松核的分布。本文在多种分析框架下对这两种分布进行了详细探讨。为提升这些分布的实际应用价值，本文提出了在数值稳定性与参数估计方面更具优势的替代参数化方法，例如采用牛顿-拉弗森算法进行参数估计，同时在回归框架中实现了更高效简化的处理流程。此外，本文引入了基于对数似然比检验的双样本位置检验，该检验旨在评估两个总体的位置参数是否可视为相等。研究建立了用于分类目的的最大似然判别分析框架，最后通过拟合球面柯西分布或泊松核分布的有限混合模型，解决了方向数据的聚类问题。通过全面的模拟研究和实际数据应用进行了实证验证，系统比较了球面柯西分布与泊松核分布的性能表现。