Likelihood ratio tests concerning the parameters of two multinomial populations are discussed. A stochastic ordering restriction is considered as a one sided alternative to equality. The one and two sample tests for equality versus stochastic ordering and stochastic ordering versus all alternatives are derived and their large sample distributions are obtained. The large sample distributions are mixtures of chi-squared distributions. The tests developed provide discrete analogues for the one sided Mann-Whitney-Wilcoxon and Kolmogorov-Smirnov tests.
"Likelihood Ratio Tests for and Against a Stochastic Ordering Between Multinomial Populations." Ann. Statist. 9 (6) 1248 - 1257, November, 1981. https://doi.org/10.1214/aos/1176345641