Generalized inverse Gaussian distribution

Generalized inverse Gaussian

Probability density function
Parameters	a > 0, b > 0, p real
Support	x > 0
pdf	$f(x) = \frac{(a/b)^{p/2}}{2 K_p(\sqrt{ab})} x^{(p-1)} e^{-(ax + b/x)/2}$
Mean	$\frac{\sqrt{b}\ K_{p+1}(\sqrt{a b}) }{ \sqrt{a}\ K_{p}(\sqrt{a b})}$
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	$\left(\frac{a}{a-2it}\right)^{\frac{p}{2}}\frac{K_p(\sqrt{b(a-2it})}{K_p(\sqrt{ab})}$

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

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

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. It is also known as the Sichel distribution, after Herbert Sichel. Its statistical properties are discussed in Bent Jørgensen's lecture notes.

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

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

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, and the hyperbolic distribution, for p=0.

Entropy

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

$H(f(x))=\frac{1}{2} \log \left(\frac{b}{a}\right)+\log \left(2 K_p\left(\sqrt{a b}\right)\right)- (p-1) \frac{\left[\frac{d}{d\nu}K_\nu\left(\sqrt{ab}\right)\right]_{\nu=p}}{K_p\left(\sqrt{a b}\right)}+\frac{\sqrt{a b}}{2 K_p\left(\sqrt{a b}\right)}\left( K_{p+1}\left(\sqrt{a b}\right) + K_{p-1}\left(\sqrt{a b}\right)\right)$

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

See also

• Inverse Gaussian distribution
• Gamma distribution