Brunk, Franck, Hanson and Hogg (1966) ("Maximum likelihood estimation of the distributions of two stochastically ordered random variables," J. Amer. Statist. Assoc. 61 1067-1080) found and studied maximum likelihood estimates of a pair of stochastically ordered distribution functions. In this paper we discuss a generalization of this problem in which we do not require the domain of these "distribution functions" to be the real line. We think of the order restriction we impose on these "distribution functions" as an analogue of stochastic ordering on the line. Maximum likelihood estimates are found and strong uniform consistency properties are discussed.
"On the Maximum Likelihood Estimation of Stochastically Ordered Random Variates." Ann. Statist. 2 (3) 528 - 534, May, 1974. https://doi.org/10.1214/aos/1176342712