The existence of an estimator constrained to lie in a certain type of bounded set is established for a fairly wide class of probability density functions. The necessary and sufficient conditions thus obtained provide a convenient means of finding such an estimator by mathematical programming methods. This result is a generalization of Cramer’s demonstration of the existence of an unconstrained maximum likelihood estimator and of Aitchison and Silvey’s demonstration of the existence of a maximum likelihood estimator constrained to satisfy certain equations.