On Stochastic Orders and Total Positivity

TL;DR

This paper reviews stochastic and likelihood ratio orders, explores their relation to total positivity of order two (TP2), and demonstrates stability under weak convergence for these distribution classes.

Contribution

It establishes the equivalence between likelihood ratio order of conditional distributions and TP2 property of joint distributions, and shows stability of these properties under weak convergence.

Findings

Likelihood ratio order of conditionals iff joint distribution is TP2

Stability of stochastic, likelihood ratio, and TP2 properties under weak convergence

Weak convergence of TP2 distributions implies convergence of conditional distributions

Abstract

The usual stochastic order and the likelihood ratio order between probability distributions on the real line are reviewed in full generality. In addition, for the distribution of a random pair $(X,Y)$, it is shown that the conditional distributions of $Y$, given $X = x$, are increasing in $x$ with respect to the likelihood ratio order if and only if the joint distribution of $(X,Y)$ is totally positive of order two (TP2) in a certain sense. It is also shown that these three types of constraints are stable under weak convergence, and that weak convergence of TP2 distributions implies convergence of the conditional distributions just mentioned.