  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. The
approximate maximum likelihood estimation (AMLE) approach is employed. It is a
general method that can be applied to mixtures with any component densities, as
long as simulation is feasible. The focus is on the dynamic
lognormal-generalized Pareto distribution, and the Cram\'er - von Mises
distance is used to measure the discrepancy between observed and simulated
samples. After deriving the theoretical properties of the estimators, a hybrid
procedure is developed, 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.
