On the existence and uniqueness of maximizers of two likelihood functions

While applying theclassical maximum likelihood method for a certain statistical inference problem, Smith and Weissman 5] have noted that there are conditions under which the likelihood function may be unbounded above or may not possess local maximizers. Ariyawansà and Templeton 1] have derived inference procedures for this problem using the theory of structural inference 2,3,4]. Based on numerical experience, and without proof, they state that the resulting likelihood functions possess unique, global maximizers, even in instances where the classical maximum likelihood method fails in the above sense. In this paper, we prove that under quite mild conditions, these likelihood functions that result from the application of the theory of structural inference are well-behaved, and possess unique, global maximizers. This research was supported in part by the Applied Mathematical Sciences subprogram of the U.S. Department of Energy under contract W-31-109-Eng-38 while the author was visiting the Mathematics and Computer Science Division of Argonne National Laboratory, Argonne, Illinois.