混合von Mises模型的参数估计

INFERENCE FOR VON MISES MIXTURE IN MEAN DIRECTION AND CONCENTRATION PARAMETERS

有限混合von Mises模型在天文学、生物学、地理和医药等许多领域都有重要的应用．可是,不论样本量有多大,此模型的似然函数都是无界的．因此,参数的最大似然估计(MLE)是不相合的．我们发现,与混合正态模型一样,上述困难可以通过引入关于分布浓度参数的一个惩罚函数或对参数空间添加适当的约束来克服．在此文中,我们从理论上证明了这两种方法是可行的,相应的参数估计是强相合的,且是渐近有效的．我们还通过计算机模拟来探讨这些新方法在有限样本情况下的统计性质,并与现有的矩估计作了比较．结果发现,惩罚极大似然估计在均方误差方面表现最佳．最后我们还分析了一组实际数据,以进一步介绍新的估计方法．

The finite mixtures of von Mises distributions in both mean direction and concentration parameters are widely used in many disciplines, including astronomy, biology, ecology, geology and medicine. It is well known that the likelihood function is unbounded for any sample size. Hence, the ordinary maximum likelihood estimator （MLE） is not consistent. Similar to normal mixtures in both mean and variance parameters, this drawback of MLE will disappear by introducing a penalty function to the log-likelihood function or putting constraints on component concentration parameters （Chen et al., 2006 and Tan et al., 2006）. In this paper, we prove that both of the penalized maximum likelihood estimator and the constrained maximum likelihood estimator are asymptotically consistent and efficient. The finite sample performance of penalized MLE and constrained MLE are compared with the moment estimator （Spur and Koutbeiy, 1991） through simulations. The PMLE is found to have the best performance in term of mean square error. A real data example is used to illustrate the proposed methods.