Approximating the distribution function of risk

We consider the multiplicative risk function, R=∏i=1pxiai, where xi's are positive random variables, independent but not identically distributed. We discuss and compare the simulated distribution of Sp=ln(R) with several asymptotic approximations. We discuss the shortcomings of Monte Carlo (MC) simulation, normal approximation, and Edgeworth expansion, and use the saddlepoint approximation to compute the cumulative distribution function (CDF) of Sp. An Edgeworth expansion and a saddlepoint approximation for the independent, but not identically distributed random variables is discussed. The accuracies of the above approximations are illustrated for computing the CDF of a hazard index for specified chemicals in consumed fish. An application considers replacement of estimated CDF in the inner loop of a two-dimensional MC strategy to study variability and uncertainty with a saddlepoint approximation.