On the mixtures of length-biased Weibull distributions for loss severity modeling

This paper introduces a new class of distributions, named length-biased Weibull mixtures, in order to deal with heavy-tailed data encountered in quantitative risk modeling. As a generalization of the Erlang mixtures with common scale parameter, our proposed class possesses attractive modeling features such as flexibility to fit various distributional shapes and weak denseness in the class of distributions for all positive random variables. In particular, the asymptotic result shows that the length-biased Weibull mixture behaves like a Weibull-tail distribution, making it more appropriate to model heavy-tailed loss severity data. A method of statistical estimation using EM algorithm is discussed, and then applied to a simulated data set and real catastrophic losses for illustration.