Abstract In this paper, a new mixture family of multivariate normal distributions, formed by mixing multivariate normal distribution and a skewed distribution, is constructed. Some properties of this family, such as characteristic function, moment generating function, and the first four moments are derived. The distributions of affine transformations and canonical forms of the model are also derived. An EM-type algorithm is developed for the maximum likelihood estimation of model parameters. Some special cases of the family, using standard gamma and standard exponential mixture distributions, denoted by MMNG and MMNE , respectively, are considered. For the proposed family of distributions, different multivariate measures of skewness are computed. In order to examine the performance of the developed estimation method, some simulation studies are carried out to show that the maximum likelihood estimates do provide a good performance. For different choices of parameters of MMNE distribution, several multivariate measures of skewness are computed and compared. Because some measures of skewness are scalar and some are vectors, in order to evaluate them properly, a simulation study is carried out to determine the power of tests, based on sample versions of skewness measures as test statistics for testing the fit of the MMNE distribution. Finally, two real data sets are used to illustrate the usefulness of the proposed model and the associated inferential methods.

中文翻译：

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多元正态分布的均值混合族：多元偏度的属性、推理和评估

摘要 本文构建了一个新的多元正态分布混合族，由多元正态分布和偏态分布混合而成。推导出该族的一些性质，如特征函数、矩生成函数和前四个矩。还导出了模型的仿射变换和规范形式的分布。EM 型算法被开发用于模型参数的最大似然估计。考虑使用标准伽玛分布和标准指数混合分布的家族的一些特殊情况，分别用 MMNG 和 MMNE 表示。对于建议的分布族，计算了不同的偏度多变量度量。为了检查开发的估计方法的性能，进行了一些模拟研究以表明最大似然估计确实提供了良好的性能。对于 MMNE 分布参数的不同选择，计算并比较了几种偏度的多元度量。由于一些偏度度量是标量而一些是向量，为了正确评估它们，基于偏度度量的样本版本作为检验 MMNE 分布拟合的检验统计量，进行了模拟研究以确定检验的功效. 最后，使用两个真实数据集来说明所提出模型和相关推理方法的有用性。计算并比较了几种偏度的多元度量。由于一些偏度度量是标量而一些是向量，为了正确评估它们，基于偏度度量的样本版本作为检验 MMNE 分布拟合的检验统计量，进行了模拟研究以确定检验的功效. 最后，使用两个真实数据集来说明所提出模型和相关推理方法的有用性。计算并比较了几种偏度的多元度量。由于一些偏度度量是标量而一些是向量，为了正确评估它们，基于偏度度量的样本版本作为检验 MMNE 分布拟合的检验统计量，进行了模拟研究以确定检验的功效. 最后，使用两个真实数据集来说明所提出模型和相关推理方法的有用性。