Top Qs
Timeline
Chat
Perspective

Generalized inverse Gaussian distribution

Family of continuous probability distributions From Wikipedia, the free encyclopedia

Generalized inverse Gaussian distribution
Remove ads

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

Quick Facts Parameters, Support ...
Remove ads

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]

Remove ads

Properties

Summarize
Perspective

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.[5]

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

Characteristic Function

The characteristic of a random variable is given as (for a derivation of the characteristic function, see supplementary materials of [6])

for where denotes the imaginary number.

Remove ads
Summarize
Perspective

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.[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]

Sichel distribution

The Sichel distribution results when the GIG is used as the mixing distribution for the Poisson parameter .[10][11]

Remove ads

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

Loading content...

See also

Loading content...
Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads