Maximum likelihood estimates with order restrictions on probabilities and odds ratios: A geometric programming approach

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