Multivariate monotone likelihood ratio and uniform conditional stochastic order

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP 2 ) and multivariate monotone likelihood ratio (MLR) for probability measures on R n These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of R n . Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP 2 .