Maximum likelihood estimation of ordered multinomial probabilities by geometric programming

We propose an efficient method to compute the maximum likelihood estimator of ordered multinomial probabilities. Using the monotonicity property of the likelihood function, we reformulate the estimation problem as a geometric program, a special type of mathematical optimization problem, which can be transformed into a convex optimization problem, and then solved globally and efficiently. We implement a numerical study to illustrate its computational merits in comparison to the m-PAV algorithm proposed by [Jewell, N.P., Kalbfleisch, J., 2004. Maximum likelihood estimation of ordered multinomial parameters. Biostatistics 5, 291-306]. We also apply our proposed method to the current status data in the above mentioned reference.