Uniform conditional variability ordering of probability distributions

Abstract

Variability orderings indicate that one probability distribution is more spread out or dispersed than another. Here variability orderings are considered that are preserved under conditioning on a common subset. One density f on the real line is said to be less than or equal to another, g, in uniform conditional variability order (UCVO) if the ratio f(x)/g(x) is unimodal with the model yielding a supremum, but f and g are not stochastically ordered. Since the unimodality is preserved under scalar multiplication, the associated conditional densities are ordered either by UCVO or by ordinary stochastic order. If f and g have equal means, then UCVO implies the standard variability ordering determined by the expectation of all convex functions. The UCVO property often can be easily checked by seeing if f(x)/g(x) is log-concave. This is illustrated in a comparison of open and closed queueing network models.