Showing papers by "Barry C. Arnold published in 1987"

Generating Ordered Families of Lorenz Curves by Strongly Unimodal Distributions

Abstract: From any strongly unimodal density on the real line, it is possible to generate a one-parameter family of Lorenz curves. The resulting families of Lorenz curves are Lorenz ordered with respect to the indexing parameter. Symmetry of the unimodal density results in the generation of symmetric Lorenz curves. A related characterization of the normal distribution is presented.

The rating of players in racquetball tournaments

Abstract: On developpe des estimations du maximum de vraisemblance et de la methode des moments pour les probabilites de gains de rallyes pour certains sports dans lesquels les points ne peuvent etre comptabilises que par le serveur

The Lorenz Order in the Space of Distribution Functions

Abstract: The graphical measure of inequality proposed by Lorenz (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 Chapter 2, a restriction that our distributions be supported on the non-negative reals and have 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 (1983) 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 (1935)), we will hold fast to our restriction that all distributions to be discussed will be supported on IR+ and will have finite means. In terms of random variables our restriction is that they be non-negative with finite expectations. We will speak interchangeably of our Lorenz (partial) order as being defined on the class of distributions (supported on IR+ with finite means) or as being defined on the class of integrable non-negative random variables.

Majorization in IRn

Abstract: As mentioned in Chapter 1, the name majorization appears first in HLP (1959). The idea had appeared earlier (HLP, 1929) although unchristened. Muirhead who dealt with Z n + (i.e. vectors of non-negative integers) already had identified the partial order defined in (1.1) (i.e. majorization). But he, when he needed to refer to it, merely called it “ordering.” Perhaps it took the insight of HLP to recognize that little of Muirhead’s work need necessarily be restricted to integers, but the key ideas including Dalton’s transfer principle were already present in Muirhead’s paper. If there was anything lacking in Muirhead’s development, it was motivation for the novel results he obtained. He did exhibit the arithmetic-geometric mean inequality as an example of his general results, but proofs of that inequality are legion. If that was the only use of his “inequalities of symmetric algebraic functions of n letters”, then they might well remain buried in the Edinburgh proceedings. HLP effectively rescued Muirhead’s work from such potential obscurity. In the present book theorems will be stated in generality comparable to that achieved by HLP and will be ascribed to those authors. Muirhead’s priority will not be repeatedly asserted. HLP restricted attention to IR n + , but the restriction to the positive orthant can and will be often dispensed with. First let us establish the relationship between majorization as defined by HLP (i.e., (1.1)) and averaging as defined by Schur.

Transformations and Their Effects

Abstract: We may most easily motivate the material in the present chapter by setting it in the context of income distributions. Income distributions which exhibit a high degree of inequality (as indicated by their Lorenz curves) are generally considered to be undesirable. Consequently, there are frequent attempts to modify observed income distributions by means of intervention in the economic process. Taxation and welfare programs are obvious examples. Essentially then, we replace the original set of incomes (or, more abstractly, a vector in IR n + ) by some function of the set of incomes. Interest centers on characterizing inequality preserving and inequality attenuating transformations. We will consider both deterministic and stochastic transformations.

Some Related Orderings

Abstract: In this chapter we consider certain partial orders defined on L (the class of non-negative random variables with positive finite expections) that are closely related to the Lorenz ordering (Chapter 3). The first, *-ordering, is often easier to deal with than Lorenz ordering, and can sometimes be used to verify Lorenz ordering which it implies. The other group of partial orderings to be discussed are those known as stochastic dominance of degree k, k=l,2,.... Degree 1 is just stochastic ordering. Degree 2 is intimately related to the Lorenz order, but distinct. Higher degree stochastic orders are most frequently encountered in economic contexts. The treatment provided here is brief.