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]
Properties
Alternative parametrization
By setting and , we can alternatively express the GIG distribution as
where is the concentration parameter while is the scaling parameter.
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
The Sichel distribution[10][11] results when the GIG is used as the mixing distribution for the
Poisson parameter .
Notes
^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.
References
^
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.
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.
^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.