Two new mixture representations for the generalized inverse Gaussian distribution and their applications

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 this article, we derive two novel mixture representations for the $\mathrm{GIG}(p, a, b)$: one that expresses the distribution as a continuous mixture of inverse Gaussians and another one that expresses it as a continuous mixture of truncated exponentials. Beyond their conceptual interest, these representations are useful for random number generation. We use the first representation to derive a geometrically ergodic Gibbs sampler whose stationary distribution is $\mathrm{GIG}(p, a, b)$, and the second one to define a recursive algorithm to generate exact independent draws from the distribution for half-integer $p$. Additionally, the second representation gives rise to a recursive algorithm for evaluating the cumulative distribution function of the $\mathrm{GIG}(p, a, b)$ for half-integer $p$. The algorithms are simple and can be easily implemented in standard programming languages.