Generalized inverse Gaussian distribution

In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function

Generalized inverse Gaussian
Probability density function
Parameters a > 0, b > 0, p real
Support x > 0
PDF
Mean

Mode
Variance
MGF
CF

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. It is also known as the Sichel distribution, after Herbert Sichel.[4] Its statistical properties are discussed in Bent Jørgensen's lecture notes.[5]

Properties

Alternative parametrization

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

where is the concentration parameter while is the scaling parameter.

Summation

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

Entropy

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

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, and the hyperbolic distribution, for p = 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]

Notes

  1. 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

  1. 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.
  2. 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).
  3. Étienne Halphen was the uncle of the mathematician Georges Henri Halphen.
  4. Sichel, H.S., Statistical valuation of diamondiferous deposits, Journal of the South African Institute of Mining and Metallurgy 1973
  5. 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.
  6. O. Barndorff-Nielsen and Christian Halgreen, Infinite Divisibility of the Hyperbolic and Generalized Inverse Gaussian Distributions, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 1977
  7. 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
  8. Dimitris Karlis, "An EM type algorithm for maximum likelihood estimation of the normal–inverse Gaussian distribution", Statistics & Probability Letters 57 (2002) 43–52.
  9. Barndorf-Nielsen, O.E., 1997. Normal Inverse Gaussian Distributions and stochastic volatility modelling. Scand. J. Statist. 24, 1–13.

See also


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