# Generalized inverse Gaussian distribution

Probability density function | |

Parameters | a > 0, b > 0, p real |
---|---|

Support | x > 0 |

Mean | |

Mode | |

Variance | |

MGF | |

CF |

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 *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.

^{[1]}

^{[2]}

^{[3]}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.

^{[4]}Its statistical properties are discussed in Bent Jørgensen's lecture notes.

^{[5]}

## Contents

## Properties[edit]

### Alternative parametrization[edit]

By setting and , we can alternatively express the GIG distribution as

where is the concentration parameter while is the scaling parameter.

### Summation[edit]

Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible.^{[6]}

### Entropy[edit]

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 evaluated at

## Related distributions[edit]

### Special cases[edit]

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 , , and . A Gamma distribution of the form

is a GIG with , , and .

Other special cases include the inverse-gamma distribution, for *a* = 0, and the hyperbolic distribution, for *p* = 0.^{[7]}

### Conjugate prior for Gaussian[edit]

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 , be GIG:

and let there be observed data points, , with normal likelihood function, conditioned on

where is the normal distribution, with mean and variance . Then the posterior for , given the data is also GIG:

where .^{[note 1]}

## Notes[edit]

**^**Due to the conjugacy, these details can be derived without solving integrals, by noting that- .

*un-normalized*GIG distribution, from which the posterior parameters can be identified.

## References[edit]

**^**Seshadri, V. (1997). "Halphen's laws". In Kotz, S.; Read, C. B.; Banks, D. L.*Encyclopedia of Statistical Sciences, Update Volume 1*. New York: Wiley. pp. 302–306.**^**Perreault, L.; Bobée, B.; Rasmussen, P. F. (1999). "Halphen Distribution System. I: Mathematical and Statistical Properties".*Journal of Hydrologic Engineering*.**4**(3): 189. doi:10.1061/(ASCE)1084-0699(1999)4:3(189).**^**Étienne Halphen was the uncle of the mathematician Georges Henri Halphen.**^**Sichel, H.S., Statistical valuation of diamondiferous deposits, Journal of the South African Institute of Mining and Metallurgy 1973**^**Jørgensen, Bent (1982).*Statistical Properties of the Generalized Inverse Gaussian Distribution*. Lecture Notes in Statistics.**9**. New York–Berlin: Springer-Verlag. ISBN 0-387-90665-7. MR 0648107.**^**O. Barndorff-Nielsen and Christian Halgreen, Infinite Divisibility of the Hyperbolic and Generalized Inverse Gaussian Distributions, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 1977- ^
^{a}^{b}Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1994),*Continuous univariate distributions. Vol. 1*, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (2nd ed.), New York: John Wiley & Sons, pp. 284–285, ISBN 978-0-471-58495-7, MR 1299979 **^**Dimitris Karlis, "An EM type algorithm for maximum likelihood estimation of the normal–inverse Gaussian distribution", Statistics & Probability Letters 57 (2002) 43–52.**^**Barndorf-Nielsen, O.E., 1997.*Normal Inverse Gaussian Distributions and stochastic volatility modelling*. Scand. J. Statist. 24, 1–13.

## See also[edit]