Fern\'andez-Dur\'an and Gregorio-Dom\'inguez (2014) defined a family of probability distributions for a vector of circular random variables by considering multiple nonnegative trigonometric sums. These distributions are highly flexible and can present numerous modes and skewness. Several operations on these multivariate distributions were translated into operations on the vector of parameters; for instance, marginalization involves calculating the eigenvectors and eigenvalues of a matrix, and independence among subsets of the vector of circular variables translates to a Kronecker product of the corresponding subsets of the vector of parameters. The derivation of marginal and conditional densities from the joint multivariate density is important when applying this model in practice to real datasets. A goodness-of-fit test based on the characteristic function and an alternative parameter estimation algorithm for high-dimensional circular data was presented and applied to a real dataset on the daily times of occurrence of maxima and minima of prices in financial markets.

翻译：Fernández-Durán和Gregorio-Domínguez（2014）通过考虑多个非负三角和，定义了一类适用于圆形随机变量向量的概率分布族。这些分布具有高度灵活性，能够呈现多种模态和偏态。针对这些多元分布的若干运算被转化为参数向量上的操作；例如，边缘化涉及计算矩阵的特征向量和特征值，而圆形变量向量子集间的独立性则转化为参数向量对应子集的Kronecker积。在实际数据集应用该模型时，从联合多元密度推导边缘密度和条件密度至关重要。本文提出了一种基于特征函数的拟合优度检验方法以及针对高维圆形数据的替代参数估计算法，并将其应用于金融市场每日价格极值发生时间的真实数据集。