A simple X-validated question on approximating P(X<Y) from two independent samples of X and Y. Which can be solved by a Monte Carlo approximation to
![\mathbb{E}[\mathbb{I}_{X<Y}]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BE%7D%5B%5Cmathbb%7BI%7D_%7BX%3CY%7D%5D&bg=000000&fg=B0B0B0&s=0&c=20201002)
using all pairs of X’s and Y’s if the samples are not too large and a subsample of this set otherwise. However, I am (still) wondering if there is a more efficient way of using both samples, for instance by comparing the empirical cdfs of through using a Mann-Whitney statistic… (ChatGPT version 3.5 first returned a nonsense solution mentioning approximating the above probability when both distributions are equal [!], before acknowledging its error.)
