Generalized inverse Gaussian distribution

Generalized inverse Gaussian

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

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 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.

Properties

Summation

Barndorff-Nielsen and Halgreen proved that the GIG distribution has infinite divisibility

Entropy

The entropy of the generalized inverse Gaussian distribution is given as

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

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 $\nu =p$

Differential equation

The pdf of the generalized inverse Gaussian distribution is a solution to the following differential equation:

$\left\{{\begin{array}{l}f(x)(x(ax-2p+2)-b)+2x^{2}f'(x)=0,\\f(1)={\frac {e^{{\frac {1}{2}}(-a-b)}\left({\frac {a}{b}}\right)^{p/2}}{2K_{p}\left({\sqrt {ab}}\right)}}\end{array}}\right\}$

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

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

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.

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:

$P(z|a,b,p)={\text{GIG}}(z|a,b,p)$

and let there be $T$ observed data points, $X=x_{1},\ldots ,x_{T}$, with normal likelihood function, conditioned on $z$:

$P(X|z,\alpha ,\beta )=\prod _{i=1}^{T}N(x_{i}|\alpha +\beta z,z)$

where $N(x|\mu ,v)$ is the normal distribution, with mean $\mu$ and variance $v$. Then the posterior for $z$, given the data is also GIG:

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

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

See also

• Inverse Gaussian distribution
• Gamma distribution