Generalized inverse Gaussian distribution

The generalized inverse Gaussian (GIG) distribution supplies the positive mixing variable for the GH family. Its normalizing integral also gives the posterior moments used by EM. A standard reference is Bent Jørgensen, Statistical Properties of the Generalized Inverse Gaussian Distribution (1982; electronic edition 2012).

Definition and normalizing integral¶

Write $Y\sim\operatorname{GIG}(p,a,b)$, where $p\in\mathbb R$ and $a,b>0$. The density on $y>0$ is

Here $K_p$ is the modified Bessel function of the second kind. Its integral representations, NIST DLMF §10.32 give the identity

We use the open interior unless a boundary is named explicitly. A normalized boundary law also exists for $a=0,b>0,p<0$ (inverse gamma), or $b=0,a>0,p>0$ (gamma). Formula (1) must then be interpreted by its limit, not by substituting zero into its Bessel ratio.

The symbol $Y$ denotes a mixing variable here. In conditioning a mixture, it is called $W$; it is independent of the standard normal noise $Z$ in that construction.

Scale and concentration parameters¶

Set $\delta=\sqrt{b/a}>0$ and $\omega=\sqrt{ab}>0$, so $a=\omega/\delta$ and $b=\omega\delta$. The source calls the latter parameter $\eta$; here $\eta$ is reserved for expectation coordinates. Then

Indeed $Y/\delta\sim\operatorname{GIG}(p,\omega,\omega)$, which verifies the power $\delta^{-p}$. More generally, for $c>0$,

This scaling identity is central to GH identifiability.

Moment generating function and moments¶

Multiplication by $e^{uy}$ changes $a$ to $a-2u$ in the normalizing integral. For $u<a/2$ this gives

At $u=a/2$ the integral is finite only if $p<0$; for $u>a/2$ it diverges. For every real $r$, multiplication by $y^r$ instead changes the Bessel order:

Thus $\mathbb E[Y]=m_1$ and $\operatorname{Var}(Y)=m_2-m_1^2$, where $m_r=\mathbb E[Y^r]$. Differentiating at $r=0$ also yields an exact special-function expression for the logarithmic moment:

This is a derivative with respect to the order, not the argument of $K$. Numerical evaluation may require quadrature or order derivatives; the absence of an elementary formula does not make the identity approximate.

Tails and moment boundaries¶

With $C=(a/b)^{p/2}/[2K_p(\sqrt{ab})]$, the two endpoint asymptotics are

The exponential cutoffs make every real power moment finite for $a,b>0$. The right tail is exponential rather than a power law; the origin is suppressed faster than any power. At the boundaries the moment domains change:

Gamma uses a shape and a rate; inverse gamma uses a shape and the coefficient of $1/y$ in its exponent. See the family tour for their normal mixtures.

Skewness and kurtosis¶

Writing $v=m_2-m_1^2$, the central moments are

The standardized skewness and excess kurtosis are

They are finite in the interior. At the inverse-gamma boundary with shape $\alpha=-p$, the excess kurtosis is

It diverges as $\alpha\downarrow4$ and is undefined for $\alpha\leq4$. It does not diverge for every inverse-gamma boundary: shapes above four retain a finite fourth moment.

Exponential-family coordinates¶

Use the same statistic order throughout this batch:

The natural parameters and their inverse map are

The interior is $\mathbb R\times(-\infty,0)^2$, and the log-partition is

Differentiation yields

The exponential-family core relates $\eta=\nabla\psi$ and $\nabla^2\psi=\operatorname{Cov}(t(Y))$ to Fisher information. The IG entry states the regularity and minimality assumptions behind these identities.

Maximum likelihood and numerical limits¶

For independent positive observations $y_1,\ldots,y_n$, set $\widehat\eta=n^{-1}\sum_i(\log y_i,y_i^{-1},y_i)^\top$. Up to a term independent of the candidate parameters, the average log likelihood is

The MLE, when attained in the interior, is

It matches the three moments in (15). Strict convexity in natural coordinates gives uniqueness of an interior optimum, but does not guarantee that one exists for every empirical moment vector. At fixed $p$, only the inverse and first moments are matched. Near boundaries, Bessel ratios and nearly dependent statistics can make inversion ill-conditioned. A numerical failure or a large parameter error alone does not prove that an MLE is nonexistent; see why not gradient descent.

Hellinger distance¶

For densities $f_1,f_2$, use the convention $H^2(f_1,f_2)=1-\int\sqrt{f_1f_2}$. Let bars denote arithmetic means of the two GIG parameter triples. Applying (2) to the geometric mean of the densities gives

Equivalently, the affinity $1-H^2$ is $\exp\{\psi((\theta^{(1)}+\theta^{(2)})/2) -[\psi(\theta^{(1)})+\psi(\theta^{(2)})]/2\}$. This compares distributions even when individual parameter coordinates are poorly conditioned. The upstream note attributes this application to Shi (2016), Generalized Hyperbolic Distributions and Related Topics.

Special cases and implementation¶

The inverse Gaussian with mean $m>0$ and shape $\lambda>0$ is exactly $\operatorname{GIG}(-1/2,\lambda/m^2,\lambda)$. Gamma and inverse gamma are the boundary cases above. Their induced marginals are described in the GH family tour.

Executable examples remain in the upstream GIG tutorial; implementation details remain in the package API and Bessel and solver design.

Source and adaptation¶

Adapted from xshi19/normix, docs/theory/gig.md, at revision 763bb3608920661a012cf089888d349fbf680aad (2026-09-13 import). Copyright (c) 2020 xshi19. Licensed under MIT. The pinned source records the original version. Notation, mathematical qualifications, and links were adapted for this site; API recipes, executable package cells, and notebook plots are omitted. No upstream benchmark execution or formal-proof verification is claimed for this adaptation.