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Mixture distributions with dynamic weights are an efficient way of modelingloss data characterized by heavy tails. However, maximum likelihood estimationof this family of models is difficult, mostly because of the need to evaluatenumerically an intractable normalizing constant. In such a setup,simulation-based estimation methods are an appealing alternative. Theapproximate maximum likelihood estimation (AMLE) approach is employed. It is ageneral method that can be applied to mixtures with any component densities, aslong as simulation is feasible. The focus is on the dynamiclognormal-generalized Pareto distribution, and the Cram\'er - von Misesdistance is used to measure the discrepancy between observed and simulatedsamples. After deriving the theoretical properties of the estimators, a hybridprocedure is developed, where standard maximum likelihood is first employed todetermine the bounds of the uniform priors required as input for AMLE.Simulation experiments and two real-data applications suggest that thisapproach yields a major improvement with respect to standard maximum likelihoodestimation.

Unsupervised Mixture Estimation via Approximate Maximum Likelihood based on the Cram\'er - von Mises distance