标题： 球面上的笛卡尔统计学 显示英文标题

标题： Cartesian Statistics on Spheres

摘要： 方向数据由q维中的单位向量组成，可以用极坐标或笛卡尔坐标描述。 轴数据可以看作是相反方向的一对方向，或者作为秩为1的投影矩阵。 从历史上看，它们的统计分析主要基于随机方向或轴的几个低阶指数族模型。 归一化常数缺乏易于处理的代数形式，阻碍了更高阶指数族在较少约束建模中的使用。 感兴趣的未知方向/轴数据分布的泛函，如方向/轴均值、离散度或分布本身。 本文概述了这些泛函的非参数估计量和引导置信集。 这些过程基于以笛卡尔坐标表示的方向/轴样本的经验分布。 还简要介绍了多个均值方向或轴之间的非参数比较、均值方向趋势的估计以及限制在指定紧子集内的q维观测分析。 显示英文摘要

摘要： Directional data consists of unit vectors in q-dimensions that can be described in polar or Cartesian coordinates. Axial data can be viewed as a pair of directions pointed in opposite directions or as a projection matrix of rank 1. Historically, their statistical analysis has largely been based on a few low-order exponential family models of distributions for random directions or axes. A lack of tractable algebraic forms for the normalizing constants has hindered the use of higher-order exponential families for less constrained modeling. Of interest are functionals of the unknown distribution of the directional/axial data, such as the directional/axial mean, dispersion, or distribution itself. This paper outlines nonparametric estimators and bootstrap confidence sets for such functionals. The procedures are based on the empirical distribution of the directional/axial sample expressed in Cartesian coordinates. Sketched as well are nonparametric comparisons among multiple mean directions or axes, estimation of trend in mean directions, and analysis of q-dimensional observations restricted to lie in a specified compact subset.