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

TL;DR

This paper investigates the primal problem of multinomial likelihood maximization under convex constraints, correcting previous flaws, and introduces the PP algorithm for solutions, with implications for empirical likelihood and related statistical methods.

Contribution

It identifies and corrects flaws in dual formulations, introduces the PP algorithm for primal solutions, and clarifies conditions where empirical and multinomial likelihoods coincide.

Findings

The PP algorithm converges to primal solutions under convex constraints.

Empirical and multinomial likelihoods can coincide under specific data conditions.

The multinomial likelihood ratio often differs from the empirical likelihood ratio.

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.