Generalized inverse Gaussian
Probability density function
Parameters	a > 0,b > 0,p real
Support	x > 0
PDF	${\displaystyle f(x)=\frac{(a/b{)}^{p/2}}{2K_p(\sqrt{ab})}{x}^{(p-1)}{e}^{-(ax+b/x)/2}}$
Mean	${\displaystyle \mathrm{E}[x]=\frac{\sqrt{b}{K}_{p+1}(\sqrt{ab})}{\sqrt{a}{K}_p(\sqrt{ab})}}$ ${\displaystyle \mathrm{E}[x^{-1}]=\frac{\sqrt{a}{K}_{p+1}(\sqrt{ab})}{\sqrt{b}{K}_p(\sqrt{ab})}-\frac{2p}{b}}$ ${\displaystyle \mathrm{E}[\ln x]=\ln \frac{\sqrt{b}}{\sqrt{a}}+\frac{\mathrm{\partial }}{\mathrm{\partial }p}\ln K_p(\sqrt{ab})}$
Mode	${\displaystyle \frac{(p-1)+\sqrt{(p-1{)}^2+ab}}{a}}$
Variance	${\displaystyle \left(\frac{b}{a}\right)\left[\frac{K_{p+2}(\sqrt{ab})}{K_p(\sqrt{ab})}-{\left(\frac{K_{p+1}(\sqrt{ab})}{K_p(\sqrt{ab})}\right)}^2\right]}$
MGF	${\displaystyle {\left(\frac{a}{a-2t}\right)}^{\frac{p}{2}}\frac{K_p(\sqrt{b(a-2t)})}{K_p(\sqrt{ab})}}$
CF	${\displaystyle {\left(\frac{a}{a-2it}\right)}^{\frac{p}{2}}\frac{K_p(\sqrt{b(a-2it)})}{K_p(\sqrt{ab})}}$

Inprobability theory andstatistics, thegeneralized inverse Gaussian distribution (GIG) is a three-parameter family of continuousprobability distributions withprobability density function

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

whereK_p is amodified Bessel function of the second kind,a > 0,b > 0 andp a real parameter. It is used extensively ingeostatistics, statistical linguistics, finance, etc. This distribution was first proposed byÉtienne Halphen.^[1][2][3] It was rediscovered and popularised byOle Barndorff-Nielsen, who called it the generalized inverse Gaussian distribution. Its statistical properties are discussed in Bent Jørgensen's lecture notes.^[4]

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.

Barndorff-Nielsen and Halgreen proved that the GIG distribution isinfinitely divisible.^[5]

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

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${\displaystyle \nu =p}$

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 theimaginary number.

Theinverse Gaussian andgamma distributions are special cases of the generalized inverse Gaussian distribution forp = −1/2 andb = 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

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

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

Other special cases include theinverse-gamma distribution, fora = 0.^[7]

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

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

and let there be${\displaystyle T}$ observed data points,${\displaystyle X=x_1,\dots ,x_T}$, 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${\displaystyle S=\sum _{i=1}^T(x_i-\alpha {)}^2}$.^{[note 1]}

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

• Inverse Gaussian distribution
• Gamma distribution

• Continuous distributions
• Exponential family distributions

• Articles with short description
• Short description is different from Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from February 2012