On the estimation of long tailed skewed distributions with actuarial applications

Very long tailed skewed distributions often arise in practice. In particular, we find that size-of-loss distributions in casualty insurance are mainly of this type. Compounding explains why many of these losses have approximate Pareto, generalized Pareto, Burr, and log-t distributions. An adaptation of the empirical mean residual life function helps the statistician select the correct model to fit in these cases. It is then discovered that minimum distance estimates, in particular that of Cramér-von Mises, and minimum chi-square estimates are extremely valuable and easy to use in the case of grouped data. Two substantial examples are given, one involving hurricane losses and the other dealing with malpractice claims.