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.
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
^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).
^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. ISBN0-387-90665-7. MR0648107.
^O. Barndorff-Nielsen and Christian Halgreen, Infinite Divisibility of the Hyperbolic and Generalized Inverse Gaussian Distributions, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 1977
^ abJohnson, 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, ISBN978-0-471-58495-7, MR1299979
^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.