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.
Publié le : 1977-05-14
Classification:
Mixtures,
random environment,
de Finetti's theorem,
positive dependence,
order statistics,
majorization,
62G30,
62E10,
62N05
@article{1176343847,
author = {Shaked, Moshe},
title = {A Concept of Positive Dependence for Exchangeable Random Variables},
journal = {Ann. Statist.},
volume = {5},
number = {1},
year = {1977},
pages = { 505-515},
language = {en},
url = {http://dml.mathdoc.fr/item/1176343847}
}
Shaked, Moshe. A Concept of Positive Dependence for Exchangeable Random Variables. Ann. Statist., Tome 5 (1977) no. 1, pp. 505-515. http://gdmltest.u-ga.fr/item/1176343847/