Properties of the generalized inverse Gaussian with applications to Monte Carlo simulation and distribution function evaluation

The generalized inverse Gaussian, denoted $\mathrm{GIG}(p, a, b)$, is a flexible family of distributions that includes the gamma, inverse gamma, and inverse Gaussian distributions as special cases. In addition to its applications in statistical modeling and its theoretical interest, the GIG often arises in computational statistics, especially in Markov chain Monte Carlo (MCMC) algorithms for posterior inference. This article introduces two mixture representations for the GIG: one that expresses the distribution as a continuous mixture of inverse Gaussians and another that reveals a recursive relationship between GIGs with different values of $p$. The former representation forms the basis for a data augmentation scheme that leads to a geometrically ergodic Gibbs sampler for the GIG. This simple Gibbs sampler, which alternates between gamma and inverse Gaussian conditional distributions, can be incorporated within an encompassing MCMC algorithm when simulation from a GIG is required. The latter representation leads to algorithms for exact, rejection-free sampling as well as CDF evaluation for the GIG with half-integer $p.$ We highlight computational examples from the literature where these new algorithms could be applied.