Fitting phase-type scale mixtures to heavy-tailed data and distributions

We consider the fitting of heavy tailed data and distributions with a special attention to distributions with a non-standard shape in the body of the distribution. To this end we consider a dense class of heavy tailed distributions introduced in Bladt et al. (Scand. Actuar. J., 573-591 2015), employing an EM algorithm for the maximum likelihood estimation of its parameters. We present methods for fitting to observed data, histograms, censored data, as well as to theoretical distributions. Numerical examples are provided with simulated data and a benchmark reinsurance dataset. Empirical examples show that the methods will in most cases adequately fit both body and tail simultaneously.