A set of k points that optimally summarize a distribution is called a set of k-principal points, which is a generalization of the mean from one point to multiple points and is useful especially for multivariate distributions. This paper discusses the estimation of principal points of multivariate distributions. First, an optimal estimator of principal points is derived for multivariate distributions of location-scale families. In particular, an optimal principal points estimator of a multivariate normal distribution is shown to be obtained by using principal points of a scaled multivariate t-distribution. We also study the case of multivariate location-scale-rotation families. Numerical examples are presented to compare the optimal estimators with maximum likelihood estimators.

中文翻译：

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位置尺度和位置尺度旋转族多元分布的最优主点估计量

对分布进行最佳总结的一组 k 个点称为一组 k 主点，它是从一个点到多个点的均值的推广，特别适用于多变量分布。本文讨论了多元分布主点的估计。首先，为位置尺度家庭的多元分布推导出主要点的最佳估计量。特别是，多元正态分布的最佳主点估计器显示为通过使用缩放的多元 t 分布的主点获得。我们还研究了多变量位置尺度旋转家庭的情况。给出了数值例子来比较最优估计量和最大似然估计量。