Maximum likelihood estimation of the logarithmic series distribution

This paper discusses the maximum likelihood estimation of the parameter of the logarithmic series distribution. The univariate case is treated in Part I, the multivariate case in Part II. A simple numerical estimation procedure is suggested using a fixed point approach. Convergence to the maximum likelihood estimator is shown. In Part III convergence rate is proven to be linear which is also demonstrated through example. In addition, comparisons with Newton’s method and the secant method in the univariate case, and with Newton’s method and the projected gradient method in the multivariate case are provided.