Invariant Directional Orderings
本文研究单变量分布的一类方向序，这些序在单调变换下保持不变，包括似然比序、风险序和随机序。通过最大不变量得到简单刻画，并展示如何用这些不变预序生成“Y2比X2更右于Y1比X1”的概念。
Statistical concepts of order permeate the theory and practice of statistics. The present paper is concerned with a large class of directional orderings of univariate distributions. (What do we mean by saying that a random variable $Y$ is larger than another random variable $X$?) Attention is restricted to preorders that are invariant under monotone transformations; this includes orderings such as monotone likelihood ratio, hazard ordering, and stochastic ordering. Simple characterizations of these orderings are obtained in terms of a maximal invariant. It is shown how such invariant preorderings can be used to generate concepts of $Y_2$ being further to the right of $X_2$ than $Y_1$ is of $X_1$.