Analysis of circular data is challenging, since the usual statistical methods are unsuitable and it is necessary to use circular periodic probabilistic models. Because some actual circular datasets exhibit asymmetry and/or multimodality, finite mixtures of symmetric circular distributions to model and fit these data have been investigated. However, it is necessary to question the predominant assumption that each component in the finite mixture model is symmetric. In this study, we consider a finite mixture model of possibly skewed circular distributions and discuss the expectation-maximization (EM) algorithm for the maximum likelihood estimate. It is shown that the maximum likelihood estimator is strongly consistent under some suitable conditions in a finite mixture of skew-symmetric circular distributions. A modified M-step in the EM algorithm is proposed in order to estimate the unknown parameter vectors effectively. To investigate the performance of our proposed model with its estimation procedure, we provide a numerical example as well as data analysis using the records of the time of day of fatal traffic accidents.

中文翻译：

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斜对称圆形分布的有限混合模型的估计

循环数据的分析具有挑战性，因为通常的统计方法不合适，需要使用循环周期概率模型。因为一些实际的圆形数据集表现出不对称性和/或多峰性，已经研究了对称圆形分布的有限混合来建模和拟合这些数据。然而，有必要质疑有限混合模型中的每个组件都是对称的主要假设。在这项研究中，我们考虑可能偏斜圆形分布的有限混合模型，并讨论最大似然估计的期望最大化 (EM) 算法。结果表明，在斜对称圆形分布的有限混合中，最大似然估计量在一些合适的条件下具有很强的一致性。为了有效地估计未知参数向量，提出了一种改进的EM算法的M步。为了研究我们提出的模型及其估计程序的性能，我们提供了一个数值示例以及使用致命交通事故当天时间记录的数据分析。