Smooth generators of integral stochastic orders. (Q1872366)

In a number of investigations it is helpful to have a rule of comparison of any two probability distributions from a given class. These rules are known as stochastic orders. Some of these stochastic orders can be characterized as integral stochastic orders. Let \(P\) and \(Q\) be probability distributions in Euclidean space \(R^d\) and \(\mathcal F\) be a class of real functions which are defined on \(R^d\). One can say that \(P\) is smaller than \(Q\) if \(\int_{\mathbb R^d}f\,dP\leq\int_{\mathbb R^d} f \,dQ\) for all \(f\in\mathcal F\). There are some problems with respect to this integral stochastic order. One can be interested in that what kind of probability distributions may be compared and what the smallest class \(\mathcal F\) may be used. The main attention is given to the second question. The authors prove that under some assumptions for \(\mathcal F\) a class of infinitely differentiable functions may be taken. This result is illustrated by a number of examples of integral stochastic orders.