Parameters Probability density function a > 0, b > 0, p real x > 0 ${\displaystyle f(x)={\frac {(a/b)^{p/2)){2K_{p}({\sqrt {ab)))))x^{(p-1)}e^{-(ax+b/x)/2))$ ${\displaystyle \operatorname {E} [x]={\frac ((\sqrt {b))\ K_{p+1}({\sqrt {ab)))}((\sqrt {a))\ K_{p}({\sqrt {ab)))))}$${\displaystyle \operatorname {E} [x^{-1}]={\frac ((\sqrt {a))\ K_{p+1}({\sqrt {ab)))}((\sqrt {b))\ K_{p}({\sqrt {ab)))))-{\frac {2p}{b))}$${\displaystyle \operatorname {E} [\ln x]=\ln {\frac {\sqrt {b)){\sqrt {a))}+{\frac {\partial }{\partial p))\ln K_{p}({\sqrt {ab)))}$ ${\displaystyle {\frac {(p-1)+{\sqrt {(p-1)^{2}+ab))}{a))}$ ${\displaystyle \left({\frac {b}{a))\right)\left[{\frac {K_{p+2}({\sqrt {ab)))}{K_{p}({\sqrt {ab)))))-\left({\frac {K_{p+1}({\sqrt {ab)))}{K_{p}({\sqrt {ab)))))\right)^{2}\right]}$ ${\displaystyle \left({\frac {a}{a-2t))\right)^{\frac {p}{2)){\frac {K_{p}({\sqrt {b(a-2t))))}{K_{p}({\sqrt {ab)))))}$ ${\displaystyle \left({\frac {a}{a-2it))\right)^{\frac {p}{2)){\frac {K_{p}({\sqrt {b(a-2it))))}{K_{p}({\sqrt {ab)))))}$

In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function

${\displaystyle f(x)={\frac {(a/b)^{p/2)){2K_{p}({\sqrt {ab)))))x^{(p-1)}e^{-(ax+b/x)/2},\qquad x>0,}$

where Kp is a modified Bessel function of the second kind, a > 0, b > 0 and p a real parameter. It is used extensively in geostatistics, statistical linguistics, finance, etc. This distribution was first proposed by Étienne Halphen.[1][2][3] It was rediscovered and popularised by Ole Barndorff-Nielsen, who called it the generalized inverse Gaussian distribution. Its statistical properties are discussed in Bent Jørgensen's lecture notes.[4]

## Properties

### Alternative parametrization

By setting ${\displaystyle \theta ={\sqrt {ab))}$ and ${\displaystyle \eta ={\sqrt {b/a))}$, we can alternatively express the GIG distribution as

${\displaystyle f(x)={\frac {1}{2\eta K_{p}(\theta )))\left({\frac {x}{\eta ))\right)^{p-1}e^{-\theta (x/\eta +\eta /x)/2},}$

where ${\displaystyle \theta }$ is the concentration parameter while ${\displaystyle \eta }$ is the scaling parameter.

### Summation

Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible.[5]

### Entropy

The entropy of the generalized inverse Gaussian distribution is given as[citation needed]

{\displaystyle {\begin{aligned}H={\frac {1}{2))\log \left({\frac {b}{a))\right)&{}+\log \left(2K_{p}\left({\sqrt {ab))\right)\right)-(p-1){\frac {\left[{\frac {d}{d\nu ))K_{\nu }\left({\sqrt {ab))\right)\right]_{\nu =p)){K_{p}\left({\sqrt {ab))\right)))\\&{}+{\frac {\sqrt {ab)){2K_{p}\left({\sqrt {ab))\right)))\left(K_{p+1}\left({\sqrt {ab))\right)+K_{p-1}\left({\sqrt {ab))\right)\right)\end{aligned))}

where ${\displaystyle \left[{\frac {d}{d\nu ))K_{\nu }\left({\sqrt {ab))\right)\right]_{\nu =p))$ is a derivative of the modified Bessel function of the second kind with respect to the order ${\displaystyle \nu }$ evaluated at ${\displaystyle \nu =p}$

### Characteristic Function

The characteristic of a random variable ${\displaystyle X\sim GIG(p,a,b)}$ is given as(for a derivation of the characteristic function, see supplementary materials of [6] )

${\displaystyle E(e^{itX})=\left({\frac {a}{a-2it))\right)^{\frac {p}{2)){\frac {K_{p}\left({\sqrt {(a-2it)b))\right)}{K_{p}\left({\sqrt {ab))\right)))}$

for ${\displaystyle t\in \mathbb {R} }$ where ${\displaystyle i}$ denotes the imaginary number.

## Related distributions

### Special cases

The inverse Gaussian and gamma distributions are special cases of the generalized inverse Gaussian distribution for p = −1/2 and b = 0, respectively.[7] Specifically, an inverse Gaussian distribution of the form

${\displaystyle f(x;\mu ,\lambda )=\left[{\frac {\lambda }{2\pi x^{3))}\right]^{1/2}\exp {\left({\frac {-\lambda (x-\mu )^{2)){2\mu ^{2}x))\right)))$

is a GIG with ${\displaystyle a=\lambda /\mu ^{2))$, ${\displaystyle b=\lambda }$, and ${\displaystyle p=-1/2}$. A Gamma distribution of the form

${\displaystyle g(x;\alpha ,\beta )=\beta ^{\alpha }{\frac {1}{\Gamma (\alpha )))x^{\alpha -1}e^{-\beta x))$

is a GIG with ${\displaystyle a=2\beta }$, ${\displaystyle b=0}$, and ${\displaystyle p=\alpha }$.

Other special cases include the inverse-gamma distribution, for a = 0.[7]

### Conjugate prior for Gaussian

The GIG distribution is conjugate to the normal distribution when serving as the mixing distribution in a normal variance-mean mixture.[8][9] Let the prior distribution for some hidden variable, say ${\displaystyle z}$, be GIG:

${\displaystyle P(z\mid a,b,p)=\operatorname {GIG} (z\mid a,b,p)}$

and let there be ${\displaystyle T}$ observed data points, ${\displaystyle X=x_{1},\ldots ,x_{T))$, with normal likelihood function, conditioned on ${\displaystyle z:}$

${\displaystyle P(X\mid z,\alpha ,\beta )=\prod _{i=1}^{T}N(x_{i}\mid \alpha +\beta z,z)}$

where ${\displaystyle N(x\mid \mu ,v)}$ is the normal distribution, with mean ${\displaystyle \mu }$ and variance ${\displaystyle v}$. Then the posterior for ${\displaystyle z}$, given the data is also GIG:

${\displaystyle P(z\mid X,a,b,p,\alpha ,\beta )={\text{GIG))\left(z\mid a+T\beta ^{2},b+S,p-{\frac {T}{2))\right)}$

where ${\displaystyle \textstyle S=\sum _{i=1}^{T}(x_{i}-\alpha )^{2))$.[note 1]

### Sichel distribution

The Sichel distribution[10][11] results when the GIG is used as the mixing distribution for the Poisson parameter ${\displaystyle \lambda }$.

## Notes

1. ^ Due to the conjugacy, these details can be derived without solving integrals, by noting that
${\displaystyle P(z\mid X,a,b,p,\alpha ,\beta )\propto P(z\mid a,b,p)P(X\mid z,\alpha ,\beta )}$.
Omitting all factors independent of ${\displaystyle z}$, the right-hand-side can be simplified to give an un-normalized GIG distribution, from which the posterior parameters can be identified.

## References

1. ^ Seshadri, V. (1997). "Halphen's laws". In Kotz, S.; Read, C. B.; Banks, D. L. (eds.). Encyclopedia of Statistical Sciences, Update Volume 1. New York: Wiley. pp. 302–306.
2. ^ Perreault, L.; Bobée, B.; Rasmussen, P. F. (1999). "Halphen Distribution System. I: Mathematical and Statistical Properties". Journal of Hydrologic Engineering. 4 (3): 189. doi:10.1061/(ASCE)1084-0699(1999)4:3(189).
3. ^ Étienne Halphen was the grandson of the mathematician Georges Henri Halphen.
4. ^ Jørgensen, Bent (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics. Vol. 9. New York–Berlin: Springer-Verlag. ISBN 0-387-90665-7. MR 0648107.
5. ^ O. Barndorff-Nielsen and Christian Halgreen, Infinite Divisibility of the Hyperbolic and Generalized Inverse Gaussian Distributions, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 1977
6. ^ Pal, Subhadip; Gaskins, Jeremy (23 May 2022). "Modified Pólya-Gamma data augmentation for Bayesian analysis of directional data". Journal of Statistical Computation and Simulation. 92 (16): 3430–3451. doi:10.1080/00949655.2022.2067853. ISSN 0094-9655. S2CID 249022546.
7. ^ a b Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1994), Continuous univariate distributions. Vol. 1, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (2nd ed.), New York: John Wiley & Sons, pp. 284–285, ISBN 978-0-471-58495-7, MR 1299979
8. ^ Dimitris Karlis, "An EM type algorithm for maximum likelihood estimation of the normal–inverse Gaussian distribution", Statistics & Probability Letters 57 (2002) 43–52.
9. ^ Barndorf-Nielsen, O.E., 1997. Normal Inverse Gaussian Distributions and stochastic volatility modelling. Scand. J. Statist. 24, 1–13.
10. ^ Sichel, Herbert S, 1975. "On a distribution law for word frequencies." Journal of the American Statistical Association 70.351a: 542-547.
11. ^ Stein, Gillian Z., Walter Zucchini, and June M. Juritz, 1987. "Parameter estimation for the Sichel distribution and its multivariate extension." Journal of the American Statistical Association 82.399: 938-944.