Lorenz ordering of means and medians
type: Item
instance of: scholarly article
title: Lorenz ordering of means and medians
zbMATH Open document ID: 0587.60017
DOI: 10.1016/0167-7152(86)90039-8
author: Barry C. Arnold
author: José A. Villaseñor
published in: Statistics \& Probability Letters
publication date: 1986-01-01T00:00:00Z
review text: The authors introduce a partial ordering \(''\leq_ L''\) on the set of nonnegative random variables, which they call the Lorenz partial order. They show that \(X\leq_ LY\) if and only if Y is a dilation of X (that is, Eh(X)\(\leq Eh(Y)\) for every convex function h for which the expectations exist). They show that \(\bar X_ n\leq_ L\bar X_{n-1}\) where, for each m, \(\bar X_ m\) denotes the sample mean of m exchangeable random variables \(X_ 1,...,X_ m.\) They also show that if \(X_ 1,...,X_{2n+3}\) are independent and identically distributed random variables with a common symmetric density then the sample medians are Lorenz ordered, that is, \(X_{n+2,2n+3}\leq_ LX_{n+1,2n+1}\) where, for each i and j (i\(\leq j)\), \(X_{i,j}\), denotes the i-th order statistic in a sample of size j of independent identically distributed random variables \(X_ 1,...,X_ j\). Finally they show that \(\bar X_{2n+1}\leq_ LX_{n+1,2n+1}\) if the parent distribution of the underlying independent identically distributed random variables is either exponential or Bernoulli or uniform.
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60G09
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 3942602
zbMATH Keywords: Lorenz partial order
zbMATH Keywords: exchangeable random variables
zbMATH Keywords: order statistic
reviewed by: Moshe Shaked
MaRDI profile type: Publication
full work available at URL: 0167-7152(86)90039-8
OpenAlex ID: W2046063546
cites work: Q4027700
cites work: Q3886649
cites work: The Existence of Probability Measures with Given Marginals
description: scientific article; zbMATH DE number 3942602