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The generalized inverse Gaussian, denoted $\mathrm{GIG}(p, a, b)$, is aflexible family of distributions that includes the gamma, inverse gamma, andinverse Gaussian distributions as special cases. In addition to itsapplications in statistical modeling and its theoretical interest, the GIGoften arises in computational statistics, especially in Markov chain MonteCarlo (MCMC) algorithms for posterior inference. This article introduces twomixture representations for the GIG: one that expresses the distribution as acontinuous mixture of inverse Gaussians and another that reveals a recursiverelationship between GIGs with different values of $p$. The formerrepresentation forms the basis for a data augmentation scheme that leads to ageometrically 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 froma 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 newalgorithms could be applied.

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