Saddlepoint approximation to gamma convolutions

Abstract

Gamma convolutions have been widely used in statistics and a wide range of scientific fields such as finance. Due to the difficulty of achieving analytically tractable expressions, the practical use of such models is limited. In this paper, we provide saddlepoint approximations of gamma convolutions employing different saddlepoint approximation methods\-Daniels' (1954) and its normalized versions for density approximation, and Lugannani and Rice's (1980) and its generalized (Wood et al., 1993) versions for cumulative distribution approximation. We examined their accuracies comparing to a simple gamma approximation recently proposed by Stewart et al. (2007) and numerically evaluated exact distribution (Sim, 1992) of gamma convolutions. Three illustrative examples demonstrate that the saddlepoint approximation techniques provide extremely accurate approximations to the exact distributions of the gamma convolutions, especially to very small tail probabilities. R programming codes for the four versions of the saddlepoint approximation are provided in Appendix.