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Stochastic orders and majorization of mean order statistics

Part of: Distribution theory 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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We characterize the (continuous) majorization of integrable functions introduced by Hardy, Littlewood, and Pólya in terms of the (discrete) majorization of finite-dimensional vectors, introduced by the same authors. The most interesting version of this result is the characterization of the (increasing) convex order for integrable random variables in terms of majorization of vectors of expected order statistics. Such a result includes, as particular cases, previous results by Barlow and Proschan and by Alzaid and Proschan, and, in a sense, completes the picture of known results on order statistics. Applications to other stochastic orders are also briefly considered.

Keywords

Order statisticstochastic orderingconvex orderincreasing convex orderdiscrete majorizationcontinuous majorizationSchur-convex function

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 60E99: None of the above, but in this section 62E99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 43 , Issue 3 , September 2006 , pp. 704 - 712
Copyright
© Applied Probability Trust 2006 

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