Quadratic distance estimators for the zeta family

The zeta distribution is a discrete distribution which has been relatively little used in actuarial science and statistics, a reason being that most estimators proposed in the literature for the parameter of this distribution require iterative methods or the extensive use of tables for its calculation, due to the complicated form of its probability mass function. In this paper, we propose a new estimator, based on quadratic distance, asymptotically fully efficient for parameter values greater than 2 and highly efficient for smaller values, but computationally more appealing than the maximum likelihood estimator; we also compare its asymptotic variance with that of the moment estimator and the estimator based on the ratio of the observed frequencies of the first two classes.