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

[1][2][3] 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.

, we can alternatively express the GIG distribution as where

is the concentration parameter while

is the scaling parameter.

Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible.

[5] The entropy of the generalized inverse Gaussian distribution is given as[citation needed] where

is a derivative of the modified Bessel function of the second kind with respect to the order

The characteristic of a random variable

is given as(for a derivation of the characteristic function, see supplementary materials of [6] ) for

denotes the imaginary number.

The inverse Gaussian and gamma distributions are special cases of the generalized inverse Gaussian distribution for p = −1/2 and b = 0, respectively.

[7] Specifically, an inverse Gaussian distribution of the form is a GIG with

A Gamma distribution of the form is a GIG with

Other special cases include the inverse-gamma distribution, for a = 0.

[7] The GIG distribution is conjugate to the normal distribution when serving as the mixing distribution in a normal variance-mean mixture.

[8][9] Let the prior distribution for some hidden variable, say

observed data points,

, with normal likelihood function, conditioned on

, given the data is also GIG: where

[note 1] The Sichel distribution results when the GIG is used as the mixing distribution for the Poisson parameter