Abstract
The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.
Highlights
Given two random variables X1 and X2 that are not necessarily independent, we have provided several results concerning the distances | X1 − X2 |, ( X1 − X2 )+ and their expectations
If X is a random variable with strictly increasing distribution function F and X1 and
An example is the median absolute deviation ν( X ) = E(| X − m X |), where m X is the median of X, which can be written in the form
Summary
Comparisons of Some Distances between Random Variables. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Different choices of Ω give rise to different stochastic orderings between | X1 − X2 | and |Y1 − Y2 | This problem was addressed in [5,6,7] for the case where X and Y have independent components with the same marginal distribution functions (see Section 2 below for details). We are concerned with two random vectors X = ( X1 , X2 ) and Y = (Y1 , Y2 ), whose components are not necessarily independent nor are they required to have identical distribution functions In this case, we explore conditions under which. In. Section 3, given a random variable X with distribution function F, we show that any functional of the form ν( X ) = E(| X1 − X2 |), where X1 and X2 are two copies of X with any type of dependence structure, is a measure of variability of X. Given any other random variable Z, we denote by FZ its distribution function
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