Journal of Applied Mathematics and Decision Sciences
Volume 1 (1997), Issue 1, Pages 53-65
Maximum likelihood estimates with order restrictions on probabilities and odds ratios: A geometric programming approach
1Dept. of Industrial Engineering, The University of Iowa, Iowa City 52242, IA, USA
2Dept. of Management Science, The University of Iowa, Iowa City 52242, IA, USA
3Milliman and Robertson, Actuaries and Consultants, St. Louis, MO, USA
Copyright © 1997 D. L. Bricker et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The problem of assigning cell probabilities to maximize a multinomial likelihood with order restrictions on the probabilies and/or restrictions on the local odds ratios is modeled as a posynomial geometric program (GP), a class of nonlinear optimization problems with a well-developed duality theory and collection of algorithms. (Local odds ratios provide a measure of association between categorical random variables.) A constrained multinomial MLE example from the literature is solved, and the quality of the solution is compared with that obtained by the iterative method of El Barmi and Dykstra, which is based upon Fenchel duality. Exploiting the proximity of the GP model of MLE problems to linear programming (LP) problems, we also describe as an alternative, in the absence of special-purpose GP software, an easily implemented successive LP approximation method for solving this class of MLE problems using one of the readily available LP solvers.