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.

翻译：广义逆高斯分布（记为$\mathrm{GIG}(p, a, b)$）是一个灵活的分布族，包含伽马分布、逆伽马分布和逆高斯分布作为特例。本文推导了$\mathrm{GIG}(p, a, b)$的两种新型混合表示：一种将其表示为逆高斯分布的连续混合，另一种将其表示为截断指数分布的连续混合。除概念上的意义外，这些表示对随机数生成具有实用价值。我们利用第一种表示构造了几何遍历的吉布斯采样器，其平稳分布为$\mathrm{GIG}(p, a, b)$；利用第二种表示定义了一种递归算法，可对半整数$p$生成精确的独立样本。此外，第二种表示还催生了一种递归算法，用于计算半整数$p$下$\mathrm{GIG}(p, a, b)$的累积分布函数。这些算法简洁易用，可方便地在标准编程语言中实现。