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An unusual stochastic order relation with some applications in sampling and epidemic theory

Part of: Distribution theory - Probability Special processes Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

Claude Lefevre  and
Philippe Picard
Claude Lefevre*
Affiliation:
Université Libre de Bruxelles
Philippe Picard*
Affiliation:
Université de Lyon 1
*
∗Postal address: Institut de Statistique, C.P. 210, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgique.
∗∗Postal address: Mathématiques Appliquées, Université de Lyon 1, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France.
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Abstract

One expects, intuitively, that the total damage caused by an epidemic increases, in a certain sense, with the infection intensity exerted by the infectives during their lifelength. The original object of the present work is to make precise in which probabilistic terms such a statement does indeed hold true, when the spread of the disease is described by a collective Reed–Frost model and the global cost is represented by the final size and severity. Surprisingly, this problem leads us to introduce an order relation for -valued random variables, unusual in the literature, based on the descending factorial moments. Further applications of the ordering occur when comparing certain sampling procedures through the number of un-sampled individuals. In particular, it is used to reinforce slightly comparison results obtained earlier for two such samplings.

Keywords

FINAL SIZE AND SEVERITYFAMILY OF POLYNOMIALSFACTORIAL MOMENTSCOMPARTMENTAL URN MODELDEATH PROCESS

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60E15: Inequalities; stochastic orderings 60K99: None of the above, but in this section
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 1 , March 1993 , pp. 63 - 81
Copyright
Copyright © Applied Probability Trust 1993 

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