Multinomial and empirical likelihood under convex constraints: directions of recession, Fenchel duality, perturbations

Abstract

The primal problem of multinomial likelihood maximization restricted to a convex closed subset of the probability simplex is studied. Contrary to widely held belief, a solution of this problem may assign a positive mass to an outcome with zero count. Related flaws in the simplified Lagrange and Fenchel dual problems, which arise because the recession directions are ignored, are identified and corrected. A solution of the primal problem can be obtained by the PP (perturbed primal) algorithm, that is, as the limit of a sequence of solutions of perturbed primal problems. The PP algorithm may be implemented by the simplified Fenchel dual. The results permit us to specify linear sets and data such that the empirical likelihood-maximizing distribution exists and is the same as the multinomial likelihood-maximizing distribution. The multinomial likelihood ratio reaches, in general, a different conclusion than the empirical likelihood ratio. Implications for minimum discrimination information, compositional data analysis, Lindsay geometry, bootstrap with auxiliary information, and Lagrange multiplier tests are discussed.