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


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

where Kp 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.

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{align} H = \frac{1}{2} \log \left( \frac b a \right) & {} +\log \left(2 K_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{a b}\right)} \\ & {} + \frac{\sqrt{a b}}{2 K_p\left(\sqrt{a b}\right)}\left( K_{p+1}\left(\sqrt{ab}\right) + K_{p-1}\left(\sqrt{a b}\right)\right) \end{align} $$

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 $$\nu=p$$

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.

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 $$\lambda$$.