Estimating the distribution of a stochastic sum of IID random variables

Abstract

Abstract Suggested is a non-parametric method and algorithm for estimating the probability distribution of a stochastic sum of independent identically distributed continuous random variables, based on combining and numerically inverting the associated empirical characteristic function (CF) derived from the observed data. This is motivated by classical problems in financial risk management, actuarial science, and hydrological modelling. This approach can be naturally generalized to more complex semi-parametric modelling and estimating approaches, e.g., by incorporating the generalized Pareto distribution fit for modelling heavy tails of the considered continuous random variables, or by considering the weighted mixture of the parametric CFs (used to incorporate the expert knowledge) and the empirical CFs (used to incorporate the knowledge based on the observed or historical data). The suggested numerical approach is based on combination of the Gil-Pelaez inversion formulae for deriving the probability distribution (PDF and CDF) from the associated CF and the trapezoidal quadrature rule used for the required numerical integration. The presented non-parametric estimation method is related to the bootstrap estimation approach, and thus, it shares similar properties. Applicability of the proposed estimation procedure is illustrated by estimating the aggregate loss distribution from the well-known Danish fire losses data.