Saddlepoint approximation for the generalized inverse Gaussian Lévy process

The generalized inverse Gaussian (GIG) Lévy process is a limit of compound Poisson processes, including the stationary gamma process and the stationary inverse Gaussian process as special cases. However, fitting the GIG Lévy process to data is computationally intractable due to the fact that the marginal distribution of the GIG Lévy process is not convolution-closed. The current work reveals that the marginal distribution of the GIG Lévy process admits a simple yet extremely accurate saddlepoint approximation. Particularly, we prove that if the order parameter of the GIG distribution is greater than or equal to −1, the marginal distribution can be approximated accurately — no need to normalize the saddlepoint density. Accordingly, maximum likelihood estimation is simple and quick, random number generation from the marginal distribution is straightforward by using Monte Carlo methods, and goodness-of-fit testing is undemanding to perform. Therefore, major numerical impediments to the application of the GIG Lévy process are removed. We demonstrate the accuracy of the saddlepoint approximation via various experimental setups.