Stochastic orderings for discrete random variables

A number of application areas of statistics make direct use of stochastic orderings. Here the special case of discrete distributions is covered. For a given partial ordering ⪯⪯ one can define the class of all ⪯⪯-order preserving functions x⪯y⇒g(x)≤g(y)x⪯y⇒g(x)≤g(y). Stochastic orderings may be defined in terms of ⪯:X⪯stY⇔EXg(X)≤EYg(Y)⪯:X⪯stY⇔EXg(X)≤EYg(Y) for all order-preserving gg. Alternatively they may be defined directly in terms of a class of functions F:X⪯stY⇔EXg(X)≤EYg(Y)F:X⪯stY⇔EXg(X)≤EYg(Y) for all f∈Ff∈F. For discrete distributions Möbius inversions plays a useful part in the theory and there are algebraic representations for the standard ordering ≤≤ for integer grids. In the general case, based on FF, the notion of a dual cone is useful. Several examples are presented.