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.

翻译：方向数据由q维空间中的单位向量组成，既可用极坐标也可用笛卡尔坐标描述。轴向数据可视为指向相反方向的一对方向，或秩为1的投影矩阵。历史上，其统计分析主要基于少数几种低阶指数族分布模型来处理随机方向或轴向。由于归一化常数缺乏易处理的代数形式，阻碍了使用更高阶指数族进行约束更少的建模。研究关注的是方向/轴向数据未知分布的函数，如方向/轴向均值、离散度或分布本身。本文概述了此类函数的非参数估计量及自助法置信集。这些方法基于用笛卡尔坐标表示的方向/轴向样本的经验分布。同时概述了多组方向或轴向均值的非参数比较、方向均值趋势的估计，以及限制在指定紧致子集内的q维观测数据分析。