Generalized inverse Gaussian distribution

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

Generalized inverse Gaussian

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

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

where K_p 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

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

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

Sichel distribution

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

See also

• Inverse Gaussian distribution
• Gamma distribution