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

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

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Article ID: iaor1998931
Country: New Zealand
Volume: 1
Issue: 1
Start Page Number: 53
End Page Number: 65
Publication Date: Jan 1997
Journal: Journal of Applied Mathematics & Decision Sciences
Authors: , ,
Keywords: probability
Abstract:

The problem of assigning cell probabilities to maximize a multinomial likelihood with order restrictions of 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.

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