Determination of the distribution of total loss from the fractional moments of its exponential

In the insurance and banking industries, as in many branches of engineering, in many instances a model can be proposed to describe either the accumulated losses or the accumulated damage or stress. Such models are of the compound type, namely random sums of independent positive random variables, and a reliable statistical model is available for both. But computing the distribution of such random sums is a hard problem that has received much attention. Here we propose a technique based on the maximum entropy method which makes use of the fractional moments of the exponential of the accumulated loss, which is available since for many models the Laplace transform can be exactly determined. Here we provide an answer to the question of how to determine distribution of a total severity when the only information that we have consists of the Laplace transform of the compound distribution. Our technique yields fairly good reconstructions from a low number of fractional moments.