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

:$f(x) = \frac x^ e^,\qquad x>0,$

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 $\theta = \sqrt$ and $\eta = \sqrt$, we can alternatively express the GIG distribution as

:$f(x) = \frac \left(\frac\right)^ e^,$

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

: $\begin H = \frac \log \left( \frac b a \right) & +\log \left(2 K_p\left(\sqrt \right)\right) - (p-1) \frac \\ & + \frac\left( K_\left(\sqrt\right) + K_\left(\sqrt\right)\right) \end$

where \left fracK_\nu\left(\sqrt\right)\right 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 )

:$E(e^) = \left(\frac\right)^ \frac$

for $t \in \mathbb$ 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'' = âˆ’1/2 and ''b'' = 0, respectively. Specifically, an inverse Gaussian distribution of the form

: f(x;\mu,\lambda) = \left frac\right \exp

is a GIG with $a = \lambda/\mu^2$, $b = \lambda$, and $p=-1/2$. A Gamma distribution of the form

: $g(x;\alpha,\beta) = \beta^\alpha \frac 1 x^ e^$
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:
:$P(z\mid a,b,p) = \operatorname(z\mid a,b,p)$
and let there be $T$ observed data points, $X=x_1,\ldots,x_T$, with normal likelihood function, conditioned on $z:$

: $P(X\mid z,\alpha,\beta) = \prod_^T N(x_i\mid\alpha+\beta z,z)$

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:
:$P(z\mid X,a,b,p,\alpha,\beta) = \text\left(z\mid a+T\beta^2,b+S,p-\frac T 2 \right)$
where $\textstyle S = \sum_^T (x_i-\alpha)^2$.Due to the conjugacy, these details can be derived without solving integrals, by noting that
:$P(z\mid X,a,b,p,\alpha,\beta)\propto P(z\mid a,b,p)P(X\mid z,\alpha,\beta)$.
Omitting all factors independent of $z$, the right-hand-side can be simplified to give an ''un-normalized'' GIG distribution, from which the posterior parameters can be identified.

Sichel distribution

The Sichel distributionStein, Gillian Z., Walter Zucchini, and June M. Juritz, 1987. "Parameter estimation for the Sichel distribution and its multivariate extension." Journal of the American Statistical Association 82.399: 938-944. results when the GIG is used as the mixing distribution for the Poisson parameter $\lambda$.

Notes

References

See also

* Inverse Gaussian distribution
*Gamma distribution

{{DEFAULTSORT:Generalized Inverse Gaussian Distribution
Continuous distributions
Exponential family distributions