Parameters Probability density function a > 0, b > 0, p real x > 0 $f(x)={\frac {(a/b)^{p/2)){2K_{p}({\sqrt {ab)))))x^{(p-1)}e^{-(ax+b/x)/2)$ $\operatorname {E} [x]={\frac ((\sqrt {b))\ K_{p+1}({\sqrt {ab)))}((\sqrt {a))\ K_{p}({\sqrt {ab)))))$ $\operatorname {E} [x^{-1}]={\frac ((\sqrt {a))\ K_{p+1}({\sqrt {ab)))}((\sqrt {b))\ K_{p}({\sqrt {ab)))))-{\frac {2p}{b))$ $\operatorname {E} [\ln x]=\ln {\frac {\sqrt {b)){\sqrt {a))}+{\frac {\partial }{\partial p))\ln K_{p}({\sqrt {ab)))$ ${\frac {(p-1)+{\sqrt {(p-1)^{2}+ab))}{a))$ $\left({\frac {b}{a))\right)\left[{\frac {K_{p+2}({\sqrt {ab)))}{K_{p}({\sqrt {ab)))))-\left({\frac {K_{p+1}({\sqrt {ab)))}{K_{p}({\sqrt {ab)))))\right)^{2}\right]$ $\left({\frac {a}{a-2t))\right)^{\frac {p}{2)){\frac {K_{p}({\sqrt {b(a-2t))))}{K_{p}({\sqrt {ab)))))$ $\left({\frac {a}{a-2it))\right)^{\frac {p}{2)){\frac {K_{p}({\sqrt {b(a-2it))))}{K_{p}({\sqrt {ab)))))$ In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function

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

## Properties

### Alternative parametrization

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

$f(x)={\frac {1}{2\eta K_{p}(\theta )))\left({\frac {x}{\eta ))\right)^{p-1}e^{-\theta (x/\eta +\eta /x)/2},$ where $\theta$ is the concentration parameter while $\eta$ is the scaling parameter.

### Summation

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

### Entropy

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

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

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

$E(e^{itX})=\left({\frac {a}{a-2it))\right)^{\frac {p}{2)){\frac {K_{p}\left({\sqrt {(a-2it)b))\right)}{K_{p}\left({\sqrt {ab))\right)))$ for $t\in \mathbb {R}$ where $i$ 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. Specifically, an inverse Gaussian distribution of the form

$f(x;\mu ,\lambda )=\left[{\frac {\lambda }{2\pi x^{3))}\right]^{1/2}\exp {\left({\frac {-\lambda (x-\mu )^{2)){2\mu ^{2}x))\right))$ is a GIG with $a=\lambda /\mu ^{2)$ , $b=\lambda$ , and $p=-1/2$ . A Gamma distribution of the form

$g(x;\alpha ,\beta )=\beta ^{\alpha }{\frac {1}{\Gamma (\alpha )))x^{\alpha -1}e^{-\beta x)$ is a GIG with $a=2\beta$ , $b=0$ , and $p=\alpha$ .

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

### 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. Let the prior distribution for some hidden variable, say $z$ , be GIG:

$P(z\mid a,b,p)=\operatorname {GIG} (z\mid a,b,p)$ and let there be $T$ observed data points, $X=x_{1},\ldots ,x_{T)$ , with normal likelihood function, conditioned on $z:$ $P(X\mid z,\alpha ,\beta )=\prod _{i=1}^{T}N(x_{i}\mid \alpha +\beta z,z)$ where $N(x\mid \mu ,v)$ is the normal distribution, with mean $\mu$ and variance $v$ . Then the posterior for $z$ , given the data is also GIG:

$P(z\mid X,a,b,p,\alpha ,\beta )={\text{GIG))\left(z\mid a+T\beta ^{2},b+S,p-{\frac {T}{2))\right)$ where $\textstyle S=\sum _{i=1}^{T}(x_{i}-\alpha )^{2)$ .[note 1]

### Sichel distribution

The Sichel distribution results when the GIG is used as the mixing distribution for the Poisson parameter $\lambda$ .

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