Elvis Wisdom: Benchmarking Saddle Point Approximations of the Gamma Distribution

Abstract: Daniels' (1954) saddle point approximation recovers the probability density from the moment generating function by using the method of steepest descent to approximate the inverse Laplace transform. Lugannani-Rice (1980) extends this method to also recover the distribution function in a similar way. This thesis uses the Lugannani-Rice method to derive an approximation for the Gamma cumulative distribution function, which coincides with Temme's (1979) approximation to the leading term. This approximation is benchmarked for accuracy and runtime against the approximation methods used in DiDonato-Morris (1986), which includes Temme. The Lugannani-Rice/Temme approximation is fast to compute and achieves high accuracy when approximating tail probabilities and when the shape parameter alpha is large. The main downside of the approximation is that it is difficult to compute higher order terms which creates an upper bound for accuracy. The main advantage is that it fills a niche by being accurate for large alpha even at moderate deviations.