Maximum likelihood estimation of cell probabilities in constrained multinomial models

We consider the problem of maximum likelihood estimation (MLE) of the cell probabilities in a multinomial model, subject to upper- and lower-bound constraints on these probabilities. We derive simple characterization of the MLE for the cases where only one-sided (upper- or lower-) bounds on all the variables are present and for the general case where two-sided bounds are present and we give simple and efficient methods for calculating the MLE. The computational complexity of our methods range from O(k) to O[k (log k)] with k being the number of cells.