Majorization and the Lorenz Order: A Brief Introduction

1987
ISBN: 978-0-387-96592-5
Verlag: Springer

Buch, Englisch, Band 43, 122 Seiten, Format (B × H): 170 mm x 244 mm, Gewicht: 242 g

Reihe: Lecture Notes in Statistics

My interest in majorization was first spurred by Ingram aIkin's proclivity for finding Schur convex functions lurking in the problem section of every issue of the American Mathematical Monthly. Later my interest in income inequality led me again to try and "really" understand Hardy, Littlewood and Polya' s contributions to the majori­ zation literature. I have found the income distribution context to be quite convenient for discussion of inequality orderings. The pre­ sent set of notes is designed for a one quarter course introducing majorization and the Lorenz order. The inequality principles of Dalton, especially the transfer or Robin Hood principle, are given appropriate prominence. Initial versions of these notes were used in graduate statistics classes taught at the Colegio de Postgraduados, Chapingo, Mexico and the University of California, Riverside. I am grateful to students in these classes for their constructive critical commentaries. My wife Carole made noble efforts to harness my free form writ­ ing and punctuation. Occasionally I was unmoved by her requests for clarification. Time will probably prove her right in these instances also. Peggy Franklin did an outstanding job of typing the manu­ script, and patiently endured requests for innumerable modifications.

Zielgruppe

Research

Autoren/Hrsg.

Arnold, Barry C.

Inhaltsverzeichnis

1 Introduction.- 2 Majorization in IR.- Exercises.- 3 The Lorenz order in the space of distribution functions.- Exercises.- 4 Transformations and their effects.- Exercises.- 5 Multivariate and stochastic majorization.- 1 Multivariate majorization.- 2 Stochastic majorization.- Exercises.- 6 Some related orderings.- 1 Star ordering.- 2 Stochastic dominance.- Exercises.- 7 Some applications.- 1 A geometric inequality of Cesaro.- 2 Matrices with prescribed characteristic roots.- 3 Variability of sample medians and means.- 4 Reliability.- 5 Genetic selection.- 6 Large interactions.- 7 Unbiased tests.- 8 Summation modulo m.- 9 Forecasting.- 10 Ecological diversity.- References.- Author index.