Unsupervised Mixture Estimation via Approximate Maximum Likelihood based on the Cramér - von Mises distance

Mixture distributions with dynamic weights are an efficient way of modeling loss data characterized by heavy tails. However, maximum likelihood estimation of this family of models is difficult, mostly because of the need to evaluate numerically an intractable normalizing constant. In such a setup, simulation-based estimation methods are an appealing alternative. We employ the approximate maximum likelihood estimation (AMLE) approach, which is general and can be applied to mixtures with any component densities, as long as simulation is feasible. We focus on the dynamic lognormal-generalized Pareto distribution, and use the Cram\'er - von Mises distance to measure the discrepancy between observed and simulated samples. After deriving the theoretical properties of the estimators, we develop a hybrid procedure, where standard maximum likelihood is first employed to determine the bounds of the uniform priors required as input for AMLE. Simulation experiments and two real-data applications suggest that this approach yields a major improvement with respect to standard maximum likelihood estimation.