Multinomial and empirical likelihood under convex constraints: directions of recession, Fenchel duality, the PP algorithm

DOI10.1214/17-EJS1294zbMATH Open1366.62109arXiv1408.5621OpenAlexW3099376389MaRDI QIDQ2362681

Yanyan Li

Publication date: 11 July 2017

Published in: Electronic Journal of Statistics (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/1408.5621

zbMATH Keywords

epi-convergenceestimating equationcontingency tableminimum discrimination informationclosed multinomial distributionEl Barmi Dykstra dualFisher likelihoodPP algorithmSmith dualzero cell frequency

Mathematics Subject Classification ID

Estimation in multivariate analysis (62H12) Optimality conditions and duality in mathematical programming (90C46) Contingency tables (62H17)

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