The Lorenz Order in the Space of Distribution Functions

Abstract

The graphical measure of inequality proposed by Lorenz (Publ Am Stat Assoc 9:209–219 (1905)) in an income inequality context is intimately related to the concept of majorization . The Lorenz curve, however, can be meaningfully used to compare arbitrary distributions rather than distributions concentrated on n points, as is the case with the majorization partial order. The Lorenz order can, thus, be thought of as a useful generalization of the majorization order. While extending our domain of definitions in one direction, to general rather than discrete distributions, we find it convenient to add a restriction which was not assumed in Chap. 2, a restriction that our distributions be supported on the non-negative reals and have positive finite expectation. In an income or wealth distribution context the restriction to non-negative incomes is often acceptable. The restriction to distributions with finite means is potentially more troublesome. Any real world (finite) population will have a (sample) distribution with finite mean. However, a commonly used approximation to real world income distributions, the Pareto distribution, only has a finite mean if the relevant shape parameter is suitably restricted. See Arnold (Pareto distributions, 2nd edn. CRC Press, Taylor & Francis Group, Boca Raton, FL (2015)) for a detailed discussion of Pareto distributions in the income modelling context. To avoid distorted Lorenz curves (as alluded to in Exercise 1 and illustrated in Wold (Metron, 12:39–58 (1935)) ), we will hold fast to our restriction that all distributions to be discussed will be supported on $\mathbb {R}^+$ and will have positive finite means. In terms of random variables our restriction is that they be non-negative with positive finite expectations. We will speak interchangeably of our Lorenz (partial) order as being defined on the class of distributions (supported on $\mathbb {R}^+$ with positive finite means) or as being defined on the class of positively integrable non-negative random variables.