In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function
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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.[4]
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.[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.[7]
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.[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]
Sichel distribution
The Sichel distribution[10][11] results when the GIG is used as the mixing distribution for the Poisson parameter .
Due to the conjugacy, these details can be derived without solving integrals, by noting that
- .
Omitting all factors independent of , the right-hand-side can be simplified to give an un-normalized GIG distribution, from which the posterior parameters can be identified.
Seshadri, V. (1997). "Halphen's laws". In Kotz, S.; Read, C. B.; Banks, D. L. (eds.). 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).
Jørgensen, Bent (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics. Vol. 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
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.
Sichel, Herbert S, 1975. "On a distribution law for word frequencies." Journal of the American Statistical Association 70.351a: 542-547.
Stein, 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.