Improved maximum likelihood estimation for the zeta distribution

We consider the small-sample properties of the maximum likelihood estimator for the shape parameter of the zeta probability distribution. Specifically, two analytical approaches to reducing the bias of this estimator are investigated—the “corrective” method of Cox and Snell (Citation1968) and the “preventive” approach of Firth (Citation1993). Both methods are effective, but the latter approach is superior and it essentially eliminates the bias of the estimator. At the same time, the percentage mean squared error is reduced significantly. Some illustrative results also show that Firth’s bias correction is superior to bootstrap or jackknife bias correction in the context of the zeta distribution. The impact of the Cox-Snell and Firth bias correction procedures is illustrated in three applications with real-life data.