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 ) . Stochastic orderings may be defined in terms of ⪯ : X ⪯ s t Y ⇔ E X g ( X ) ≤ E Y g ( Y ) for all order-preserving g . Alternatively they may be defined directly in terms of a class of functions F : X ⪯ s t Y ⇔ E X g ( X ) ≤ E Y g ( Y ) for all f ∈ 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 F , the notion of a dual cone is useful. Several examples are presented.