Stochastic orders generated by integrals: A unified study

We consider stochastic orders of the following type. Let F be a class of functions and let P and Q be probability measures. Then define P less than or equal to(F) Q, if integral f dP less than or equal to integral f dQ for all f in F. 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.