On a Comparison between Lévy’s Inversion Formula and Saddlepoint Approximations

Abstract

The saddlepoint approximations introduced by Daniels (1954) has been used as a simple and useful way for inverting the characteristic function to corresponding p.d.f. However, nowadays the ability of the computing environment enables us to calculate the integration such as Lévy’s inversion formula that includes complex numbers. We shall compare the saddlepoint method with the formula by through some numerical examples. Although the accuracy of the former is inferior to the latter, saddlepoint itself has an inversion formula to get exact p.d.f. Moreover, it naturally explains the performance of the asymptotic normality of a statistics (Takeuchi (2013, 2014)). This is demonstrated when we evaluate the asymptotic normality of the kernel-type density estimator at each point.