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Inverse Stochastic Dominance, Majorization, and Mean Order Statistics

Part of: Inequalities Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Jesús De La Cal  and
Javier Cárcamo
Jesús De La Cal*
Affiliation:
Universidad del País Vasco
Javier Cárcamo*
Affiliation:
Universidad Autónoma de Madrid
*
∗Postal address: Departamento de Matemática Aplicada y Estadística e Investigación Operativa, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain. Email address: jesus.delacal@ehu.es
∗∗Postal address: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain. Email address: javier.carcamo@uam.es
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Abstract

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The inverse stochastic dominance of degree r is a stochastic order of interest in several branches of economics. We discuss it in depth, the central point being the characterization in terms of the weak r-majorization of the vectors of expected order statistics. The weak r-majorization (a notion introduced in the paper) is a natural extension of the classical (reverse) weak majorization of Hardy, Littlewood and Pòlya. This work also shows the equivalence between the continuous majorization (of higher order) and the discrete r-majorization. In particular, our results make it clear that the cases r = 1, 2 differ substantially from those with r ≥ 3, a fact observed earlier by Muliere and Scarsini (1989), among other authors. Motivated by this fact, we introduce new stochastic orderings, as well as new social inequality indices to compare the distribution of the wealth in two populations, which could be considered as natural extensions of the first two dominance rules and the S-Gini indices, respectively.

Keywords

Stochastic ordersinverse stochastic dominanceorder statisticmajorizationS-Gini indexinequality measurement

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 60E99: None of the above, but in this section 26D15: Inequalities for sums, series and integrals
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 277 - 292
Copyright
Copyright © Applied Probability Trust 2010 

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