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Stochastic Orders Generated by Integrals: a Unified Study
- Part of
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- Journal:
- Advances in Applied Probability / Volume 29 / Issue 2 / June 1997
- Published online by Cambridge University Press:
- 01 July 2016, pp. 414-428
- Print publication:
- June 1997
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- Article
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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.
be a class of functions and let 
, if ∫ ⨍ 
. 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 