Robust estimation of location and concentration parameters for the von Mises-Fisher distribution is discussed. A key reparametrisation is achieved by expressing the two parameters as one vector on the Euclidean space. With this representation, we first show that maximum likelihood estimator for the von Mises-Fisher distribution is not robust in some situations. Then we propose two families of robust estimators which can be derived as minimisers of two density power divergences. The presented families enable us to estimate both location and concentration parameters simultaneously. Some properties of the estimators are explored. Simple iterative algorithms are suggested to find the estimates numerically. A comparison with the existing robust estimators is given as well as discussion on difference and similarity between the two proposed estimators. A simulation study is made to evaluate finite sample performance of the estimators. We consider a sea star dataset and discuss the selection of the tuning parameters and outlier detection.
Penalized Maximum Likelihood Estimator for Mixture of von Mises-Fisher Distributions
von Mises-Fisher分布是描述方向数据的最广泛使用的概率分布之一。冯·米塞斯-费舍尔分布的有限混合物已经找到了许多应用。然而，冯Mises-Fisher分布的有限混合的似然函数是无界的，因此极大似然估计没有很好地定义。为了解决问题的可能性退化，我们考虑惩罚的极大似然的方法，其中的惩罚函数被纳入。我们证明了强一致性的估计。提出了一种罚似然函数的期望最大化算法，并进行了仿真研究。
Divergence-based Robust Generalised Bayesian Inference for Directional Data via von Mises-Fisher models
School of Data Science, Meisei University（明成大学数据科学学院）；RIKEN Center for Brain Science（理化学研究所脑科学中心）；School of Business Administration, Meiji University（明治大学商学院）；Department of Information Sciences, Tokyo University of Science（东京科学技术大学信息科学系）
