On maximum likelihood estimation of a Pareto mixture

In this paper we deal with maximum likelihood estimation (MLE) of the parameters of a Pareto mixture. Standard MLE procedures are difficult to apply in this setup, because the distributions of the observations do not have common support. We study the properties of the estimators under different hypotheses; in particular, we show that, when all the parameters are unknown, the estimators can be found maximizing the profile likelihood function. Then we turn to the computational aspects of the problem, and develop three alternative procedures: an EM-type algorithm, a Simulated Annealing and an algorithm based on Cross-Entropy minimization. The work is motivated by an application in the operational risk measurement field: we fit a Pareto mixture to operational losses recorded by a bank in two different business lines. Under the assumption that each population follows a Pareto distribution, the appropriate model is a mixture of Pareto distributions where all the parameters have to be estimated.