Maximum likelihood estimation for constrained parameters of multinomial distributions - Application to Zipf-Mandelbrot models

A numerical maximum likelihood (ML) estimation procedure is developed for the constrained parameters of multinomial distributions. The main dif��?culty involved in computing the likelihood function is the precise and fast determination of the multinomial coef��?cients. For this the coef��?cients are rewritten into a telescopic product. The presented method is applied to the ML estimation of the Zipf–Mandelbrot (ZM) distribution, which provides a true model in many real-life cases. The examples discussed arise from ecological and medical observations. Based on the estimates, the hypothesis that the data is ZM distributed is tested using a chi-square test. The computer code of the presented procedure is available on request by the author.