Generalized inverse Gaussian distribution
|Probability density function
|Parameters||a > 0, b > 0, p real|
|Support||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. It is also known as the Sichel distribution, after Herbert Sichel. Its statistical properties are discussed in Bent Jørgensen's lecture notes.
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
is a GIG with , , and . A Gamma distribution of the form
is a GIG with , , and .
The entropy of the generalized inverse Gaussian distribution is given as
where is a derivative of the modified Bessel function of the second kind with respect to the order evaluated at
- 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.
- 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. MR0648107.
- 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