Asymptotic properties of maximum likelihood estimators for a generalized Pareto-type distribution

Astola and Danielian [1], using stochastic birth-death process, have proposed a regular four-parameter discrete probability distribution, called generalized Pareto-type model, which is an appealing distribution for modeling phenomena in Bioinformatics. Farbod and Gasparian [5], fitted this distribution to the two sets of real data, and have derived conditions under which a solution for the system of likelihood equations exists and coincides with the maximum likelihood estimators (MLE) for the model unknown parameters. Also, in [5], an accumulation method for approximate computation of the MLE has been considered with simulation studies. In this paper we show that for sufficiently large sample size the system of likelihood equations has a solution, which according to [5], coincides with the MLE of vector-valued parameter for the underlying model. Besides, we establish asymptotic unbiasedness, weak consistency, asymptotic normality, asymptotic efficiency, and convergence of arbitrary moments of the MLE, by verifying the so-called regularity-conditions.