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Stochastic Orders Generated by Integrals: a Unified Study

Part of: Probability theory on algebraic and topological structures Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Alfred Müller
Alfred Müller*
Affiliation:
Universität Karlsruhe
*
∗ Postal address: Institut für Wirtschaftstheorie und Operations Research, Universität Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany.
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Abstract

We consider stochastic orders of the following type. Let be a class of functions and let P and Q be probability measures. Then define , if ∫ ⨍ d P ≦ ∫ ⨍ d Q for all f in . Marshall (1991) posed the problem of characterizing the maximal cone of functions generating such an ordering. We solve this problem by using methods from functional analysis. Another purpose of this paper is to derive properties of such integral stochastic orders from conditions satisfied by the generating class of functions. The results are illustrated by several examples. Moreover, we show that the likelihood ratio order is closed with respect to weak convergence, though it is not generated by integrals.

Keywords

INTEGRAL STOCHASTIC ORDERSCLOSURE PROPERTIESMAXIMAL GENERATORLIKELIHOOD RATIO ORDER

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60B11: Probability theory on linear topological spaces 60E05: Distributions

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 2 , June 1997 , pp. 414 - 428
Copyright
Copyright © Applied Probability Trust 1997 

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