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.
翻译:此外,对于随机分配一对(X,Y)美元的情况,还表明,如果而且只有在(X,Y)美元联合分配在某种意义上完全符合第二(TP2)号命令(TP2)时,按实际线上概率分布的概率顺序和可能性比例顺序将完全加以审查;此外,就随机分配一对美元(X,Y)而言,有条件分配的美元(X美元=x美元)正以x美元增加,如果而且只有在(X,Y)美元联合分配在某种意义上完全符合第二(TP2)号命令(TP2)的情况下,则按可能性顺序分配的金额可能以美元增加,此外,还表明这三类制约在趋同性下保持稳定,TP2号分配的薄弱趋同意味着刚刚提到的有条件分配的趋同。