General convex stochastic orderings and related martingale-type structures

Abstract

Blackwell (1951), in his seminal work on comparison of experiments, ordered two experiments using a dilation ordering: one experiment, Y, is ‘more spread out’ in the sense of dilation than another one, X, if E(c(Y))≥E(c(X)) for all convex functions c. He showed that this ordering is equivalent to two other orderings, namely (i) a total time on test ordering and (ii) a martingale relationship E(Yʹ | Xʹ)=Xʹ, where (Xʹ,Yʹ) has a joint distribution with the same marginals as X and Y. These comparisons are generalized to balayage orderings that are defined in terms of generalized convex functions. These balayage orderings are equivalent to (i) iterated total integral of survival orderings and (ii) martingale-type orderings which we refer to as k-mart orderings. These comparisons can arise naturally in model fitting and data confidentiality contexts.