Generalized inverse Gaussian distribution

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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

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Generalized inverse Gaussian
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Parameters a > 0, b > 0, p real
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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

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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.

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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]

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

See also

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