Statistical Depth Function Random Variables for Univariate Distributions and induced Divergences

2023/04/25 by Ding, Rui
#62B10 #62G05 #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2304.13091

Abstract

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.

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