Generalized inverse Gaussian distribution

Short description: Family of continuous probability distributions

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	$\mathrm{E}[x]=\frac{\sqrt{b}{K}_{p+1}(\sqrt{ab})}{\sqrt{a}{K}_p(\sqrt{ab})}$ $\mathrm{E}[x^{-1}]=\frac{\sqrt{a}{K}_{p+1}(\sqrt{ab})}{\sqrt{b}{K}_p(\sqrt{ab})}-\frac{2p}{b}$ $\mathrm{E}[\ln x]=\ln \frac{\sqrt{b}}{\sqrt{a}}+\frac{\partial }{\partial p}\ln K_p(\sqrt{ab})$
Mode	$\frac{(p-1)+\sqrt{(p-1{)}^2+ab}}{a}$
Variance	$\left(\frac{b}{a}\right)[\frac{K_{p+2}(\sqrt{ab})}{K_p(\sqrt{ab})}-{\left(\frac{K_{p+1}(\sqrt{ab})}{K_p(\sqrt{ab})}\right)}^2]$
MGF	${\left(\frac{a}{a-2t}\right)}^{\frac{p}{2}}\frac{K_p(\sqrt{b(a-2t)})}{K_p(\sqrt{ab})}$
CF	${\left(\frac{a}{a-2it}\right)}^{\frac{p}{2}}\frac{K_p(\sqrt{b(a-2it)})}{K_p(\sqrt{ab})}$

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

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

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

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

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

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

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:

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

and let there be $T$ observed data points, $X=x_1,\dots ,x_T$, 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:

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

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

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