Estimation of weighted multinomial probabilities under log-convex constraints

Many problems in Statistics involve maximizing a multinomial likelihood over a restricted region. In this paper, we consider instead maximizing a weighted multinomial likelihood. We show that a dual problem always exits which is frequently more tractable and that a solution to the dual problem leads directly to a solution of the primal problem. Moreover, the form of the dual problem suggests an iterative algorithm for solving the MLE problem when the constraint region can be written as a finite intersection of cones. We show that this iterative algorithm is guaranteed to converge to the true solution and show that when the cones are isotonic, this algorithm is a version of Dykstra's algorithm (Dykstra, J. Amer. Statist. Assoc. 78 (1983) 837–842) for the special case of least squares projection onto the intersection of isotonic cones. We give several meaningful examples to illustrate our results. In particular, we obtain the nonparametric maximum likelihood estimator of a monotone density function in the presence of selection bias.