This paper is concerned with maximum likelihood estimates for a large class of families of unimodal densities. The existence of measurable maximum likelihood estimates and the consistency of asymptotic maximum likelihood estimates are proved. By counterexamples it is shown that the conditions which are sufficient for consistency cannot be removed without compensation.
"On the Measurability and Consistency of Maximum Likelihood Estimates for Unimodal Densities." Ann. Statist. 1 (5) 888 - 901, September, 1973. https://doi.org/10.1214/aos/1176342509