LTD and RTI dependence orderings

Abstract

The authors show how the approach of Capéraà and Genest (Canad. J. Statist., 1990) can be used to order bivariate distributions with arbitrary marginals by their degree of dependence in the LTD (left-tail decreasing) or RTI (right-tail increasing) sense. Some properties of these new orderings are given, along with applications to Archimedean copulas, order statistics and compound random variables. RÉSUMÉ Les auteurs montrent comment l'approche de Capéraà et Genest (Revue canad. statist., 1990) peutêtre employée pour ordonner des lois bivariéesà marges quelconques selon leur force de dépendance au sens LTD (décroissance par ailes gauches) ou au sens RTI (croissance par ailes droites). Ils présentent quelques-unes des propriétés de ces ordres et en montrent l'intérêt pour l'étude des copules archimédiennes, des statistiques d'ordre et des sommes aléatoires.

On the dependence structure of order statistics

“…This is in contrast with the results of Ave´rous and Dortet-Bernadet [1] concerning the ordering of the largest order statistic on the smallest one in the non-copula-based formulation of the more stochastically increasing ordering that they use.…”

Some notions of multivariate positive dependence

“…Colangelo et al (2005) compared these orders to other stochastic orders, studied by Avérous and Dortet-Bernadet (2000) and by Hollander et al (1990), which generalize the same positive dependence notions, and showed that no relationship exists between them.…”

Archimedean copulae and positive dependence

“…A sufficient condition for supermodular ordering of Archimedean copulae (and thus also for supermodular positive dependence) is given in [34], see also [10,16]. Some results for the bivariate case can also be found in [6,7] and in [1]. Bassan and Spizzichinao [5] study dependence and aging concepts for bivariate distributions in terms of an object that they call Archimedean semi-copula.…”