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May, 1977 A Concept of Positive Dependence for Exchangeable Random Variables
Moshe Shaked
Ann. Statist. 5(3): 505-515 (May, 1977). DOI: 10.1214/aos/1176343847


An $n$-variate distribution function is said to be positive dependent by mixture (PDM) if it is a mixture of independent $n$-variate distributions with equal marginals. PDM distributions arise in various contexts of reliability and other areas of statistics. We give a necessary and sufficient condition, by means of independent random variables, for an $n$-variate distribution function to be PDM. The distributions and the expectations of the order statistics of PDM and of independent $n$-variate distributions which have the same marginals, are compared and the results applied to obtain bounds for the reliability of certain "$k$ out of $n$" systems. A characterization of vectors of expectations of order statistics of PDM distribution is shown. Surprisingly many exchangeable distributions are found to be PDM. We prove a closure property of the class of PDM distributions and list some examples.


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Moshe Shaked. "A Concept of Positive Dependence for Exchangeable Random Variables." Ann. Statist. 5 (3) 505 - 515, May, 1977.


Published: May, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0363.62036
MathSciNet: MR436414
Digital Object Identifier: 10.1214/aos/1176343847

Primary: 62G30
Secondary: 62E10 , 62N05

Keywords: De Finetti's theorem , majorization , mixtures , order statistics , Positive dependence , random environment

Rights: Copyright © 1977 Institute of Mathematical Statistics


Vol.5 • No. 3 • May, 1977
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