  A non-parametric method for evaluation of the aggregate loss distribution
(ALD) by combining and numerically inverting the empirical characteristic
functions (CFs) is presented and illustrated. This approach to evaluate ALD is
based on purely non-parametric considerations, i.e., based on the empirical CFs
of frequency and severity of the claims in the actuarial risk applications.
This approach can be, however, naturally generalized to a more complex
semi-parametric modeling approach, e.g., by incorporating the generalized
Pareto distribution fit of the severity distribution heavy tails, and/or by
considering the weighted mixture of the parametric CFs (used to model the
expert knowledge) and the empirical CFs (used to incorporate the knowledge
based on the historical data - internal and/or external). Here we present a
simple and yet efficient method and algorithms for numerical inversion of the
CF, suitable for evaluation of the ALDs and the associated measures of interest
important for applications, as, e.g., the value at risk (VaR). The presented
approach is based on combination of the Gil-Pelaez inversion formulae for
deriving the probability distribution (PDF and CDF) from the compound
(empirical) CF and the trapezoidal rule used for numerical integration. The
applicability of the suggested approach is illustrated by analysis of a well
know insurance dataset, the Danish fire loss data.
