Statistical analysis of the Lognormal-Pareto distribution using Probability Weighted Moments and Maximum Likelihood

This paper deals with the estimation of the lognormal-Pareto and the lognormal-Generalized Pareto mixture distributions. The log-likelihood function is discontinuous, so that Maximum Likelihood Estimation is not asymptotically optimal. For this reason, we develop an alternative method based on Probability Weighted Moments. We show that the standard version of the method can be applied to the first distribution, but not to the latter. Thus, in the lognormal- Generalized Pareto case, we work out the details of a mixed approach combining Maximum Likelihood Estimation and Probability Weighted Moments. Extensive simulations give precise indications about the relative efficiencies of the methods in various setups. Finally, we apply the techniques to two real datasets in the actuarial and operational risk management fields.