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

Victor Peña, Michael Jauch

TL;DR

This paper addresses efficient simulation and CDF evaluation for the generalized inverse Gaussian distribution GIG$(p,a,b)$. It introduces two mixture representations: a continuous mixture of inverse Gaussian distributions that supports a data-augmentation Gibbs sampler, and a mixture/recurrence representation that enables exact, rejection-free sampling and CDF evaluation for half-integer $p$. The data-augmented Gibbs sampler is shown to be geometrically ergodic, and the half-integer results yield practical algorithms with competitive performance compared to existing methods. The methods are designed to complement, not replace, existing samplers and are applicable in Bayesian posterior inference and other computational statistics tasks; supplementary material provides proofs, pseudocode, and numerical experiments.

Abstract

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.

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

TL;DR

This paper addresses efficient simulation and CDF evaluation for the generalized inverse Gaussian distribution GIG. It introduces two mixture representations: a continuous mixture of inverse Gaussian distributions that supports a data-augmentation Gibbs sampler, and a mixture/recurrence representation that enables exact, rejection-free sampling and CDF evaluation for half-integer . The data-augmented Gibbs sampler is shown to be geometrically ergodic, and the half-integer results yield practical algorithms with competitive performance compared to existing methods. The methods are designed to complement, not replace, existing samplers and are applicable in Bayesian posterior inference and other computational statistics tasks; supplementary material provides proofs, pseudocode, and numerical experiments.

Abstract

The generalized inverse Gaussian, denoted , 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 . 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 We highlight computational examples from the literature where these new algorithms could be applied.
Paper Structure (18 sections, 10 theorems, 86 equations, 2 tables, 3 algorithms)

This paper contains 18 sections, 10 theorems, 86 equations, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Data-augmented Gibbs sampler
  4. Exact sampling and CDF evaluation for half-integer $p$
  5. Conclusions
  6. Appendix
  7. Proofs of Propositions
  8. Algorithms
  9. Reparametrization of the GIG
  10. Reparametrization
  11. Propositions in the new parametrization
  12. Gibbs sampler based on Zhang and Reiter (2022)
  13. Notation for Truncated Distributions
  14. Joint Distribution
  15. Full Conditional Distributions
  16. ...and 3 more sections

Key Result

Proposition 1

The PDF of the $\mathrm{GIG}(p, a, b)$ for $p \neq -1/2$ can be written as a continuous mixture of inverse Gaussian distributions: where $f_X(x)$ is the PDF of the $\mathrm{GIG}(p, a, b)$, $f_Y(y)$ is a PDF with support on $\mathbb{R}_+$, and $f_{X \mid Y}(x \mid y)$ is an inverse Gaussian PDF. The parameters of the distributions are different for the cases $p < -1/2$ and $p > -1/2$. If $p < -1/2

Theorems & Definitions (15)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 5 more