To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Generalized inverse Gaussian distribution

From Wikipedia, the free encyclopedia

Generalized inverse Gaussian
Probability density function
Probability density plots of GIG distributions
Parametersa > 0, b > 0, p real
Supportx > 0
PDF
Mean

Mode
Variance
MGF
CF

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

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]

YouTube Encyclopedic

  • 1/5
    Views:
    12 747
    8 026
    14 839
    2 446
    1 711
  • ✪ Probability functions: pdf, CDF and inverse CDF (FRM T2-1)
  • ✪ Binomial Distribution 2: Canonical Link function
  • ✪ Mod-01 Lec-07 Normal Density and Discriminant Function
  • ✪ 09 - Exponential Family
  • ✪ Exponential Family of Distributions

Transcription

Contents

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[citation needed]

where is a derivative of the modified Bessel function of the second kind with respect to the order evaluated at

Related distributions

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. 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. ^ a b 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


This page was last edited on 16 January 2019, at 18:52
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.