In this paper, we show that the halfspace depth random variable for samples from a univariate distribution with a notion of center is distributed as a uniform distribution on the interval [0,1/2]. The simplicial depth random variable has a distribution that first-order stochastic dominates that of the halfspace depth random variable and relates to a Beta distribution. Depth-induced divergences between two univariate distributions can be defined using divergences on the distributions for the statistical depth random variables in-between these two distributions. We discuss the properties of such induced divergences, particularly the depth-induced TVD distance based on halfspace or simplicial depth functions, and how empirical two-sample estimators benefit from such transformations.

翻译：本文证明，对于具有中心化概念的单变量分布样本，半空间深度随机变量服从区间[0,1/2]上的均匀分布。单纯形深度随机变量的分布一阶随机占优半空间深度随机变量的分布，且与Beta分布相关。通过利用介于两个分布之间的统计深度随机变量的分布散度，可以定义两个单变量分布之间的深度诱导散度。我们讨论了此类诱导散度的性质，特别是基于半空间或单纯形深度函数的深度诱导全变差距离（TVD），并阐述了经验双样本估计量如何从这类变换中获益。