  Here we present an application of two maxentropic procedures to determine the
probability density distribution of compound sums of random variables, using
only a finite number of empirically determined fractional moments. The two
methods are the Standard method of Maximum Entropy (SME), and the method of
Maximum Entropy in the Mean (MEM). We shall verify that the reconstructions
obtained satisfy a variety of statistical quality criteria, and provide good
estimations of VaR and TVaR, which are important measures for risk management
purposes. We analyze the performance and robustness of these two procedures in
several numerical examples, in which the frequency of losses is Poisson and the
individual losses are lognormal random variables. As side product of the work,
we obtain a rather accurate description of the density of the compound random
variable. This is an extension of a previous application based on the Standard
Maximum Entropy approach (SME) where the analytic form of the Laplace transform
was available to a case in which only observed or simulated data is used. These
approaches are also used to develop a procedure to determine the distribution
of the individual losses through the knowledge of the total loss. Then, in the
case of having only historical total losses, it is possible to decompound or
disaggregate the random sums in its frequency/severity distributions, through a
probabilistic inverse problem.
