Fractional moment estimation of Linnik and Mittag–Leffler parameters

It is shown that Linnik and Mittag–Leffler distributions are geometric stable (GS). An estimation method for the parameters of Linnik and Mittag–Leffler distributions, based on fractional moments, is proposed. Algorithms for simulation of Linnik and Mittag–Leffler distributions, based on their representations as mixtures of Laplace and exponential distributions, are presented. The estimation procedure is validated on simulated data, and its potential data, and its potential applications are illustrated with S&P index data, that is shown to be consistent with a two-parameter Linnik model.