Generalized inverse Gaussian distribution

Generalized inverse Gaussian

Probability density function
Parameters	a > 0, b > 0, p real
Support	x > 0
PDF	${\displaystyle f(x)={\frac {(a/b)^{p/2}}{2K_{p}({\sqrt {ab}})}}x^{(p-1)}e^{-(ax+b/x)/2}}$
Mean	${\displaystyle \operatorname {E} [x]={\frac {{\sqrt {b}}\ K_{p+1}({\sqrt {ab}})}{{\sqrt {a}}\ K_{p}({\sqrt {ab}})}}}$ ${\displaystyle \operatorname {E} [x^{-1}]={\frac {{\sqrt {a}}\ K_{p+1}({\sqrt {ab}})}{{\sqrt {b}}\ K_{p}({\sqrt {ab}})}}-{\frac {2p}{b}}}$ ${\displaystyle \operatorname {E} [\ln x]=\ln {\frac {\sqrt {b}}{\sqrt {a}}}+{\frac {\partial }{\partial p}}\ln K_{p}({\sqrt {ab}})}$
Mode	${\displaystyle {\frac {(p-1)+{\sqrt {(p-1)^{2}+ab}}}{a}}}$
Variance	${\displaystyle \left({\frac {b}{a}}\right)\left[{\frac {K_{p+2}({\sqrt {ab}})}{K_{p}({\sqrt {ab}})}}-\left({\frac {K_{p+1}({\sqrt {ab}})}{K_{p}({\sqrt {ab}})}}\right)^{2}\right]}$
MGF	${\displaystyle \left({\frac {a}{a-2t}}\right)^{\frac {p}{2}}{\frac {K_{p}({\sqrt {b(a-2t)}})}{K_{p}({\sqrt {ab}})}}}$
CF	${\displaystyle \left({\frac {a}{a-2it}}\right)^{\frac {p}{2}}{\frac {K_{p}({\sqrt {b(a-2it)}})}{K_{p}({\sqrt {ab}})}}}$

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

${\displaystyle f(x)={\frac {(a/b)^{p/2}}{2K_{p}({\sqrt {ab}})}}x^{(p-1)}e^{-(ax+b/x)/2},\qquad x>0,}$

where K_p 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 ${\displaystyle \theta ={\sqrt {ab}}}$ and ${\displaystyle \eta ={\sqrt {b/a}}}$, we can alternatively express the GIG distribution as

${\displaystyle f(x)={\frac {1}{2\eta K_{p}(\theta )}}\left({\frac {x}{\eta }}\right)^{p-1}e^{-\theta (x/\eta +\eta /x)/2},}$

where ${\displaystyle \theta }$ is the concentration parameter while ${\displaystyle \eta }$ 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]}

${\displaystyle {\begin{aligned}H={\frac {1}{2}}\log \left({\frac {b}{a}}\right)&{}+\log \left(2K_{p}\left({\sqrt {ab}}\right)\right)-(p-1){\frac {\left[{\frac {d}{d\nu }}K_{\nu }\left({\sqrt {ab}}\right)\right]_{\nu =p}}{K_{p}\left({\sqrt {ab}}\right)}}\\&{}+{\frac {\sqrt {ab}}{2K_{p}\left({\sqrt {ab}}\right)}}\left(K_{p+1}\left({\sqrt {ab}}\right)+K_{p-1}\left({\sqrt {ab}}\right)\right)\end{aligned}}}$

where ${\displaystyle \left[{\frac {d}{d\nu }}K_{\nu }\left({\sqrt {ab}}\right)\right]_{\nu =p}}$ is a derivative of the modified Bessel function of the second kind with respect to the order ${\displaystyle \nu }$ evaluated at ${\displaystyle \nu =p}$

Characteristic Function

The characteristic of a random variable ${\displaystyle X\sim GIG(p,a,b)}$ is given as(for a derivation of the characteristic function, see supplementary materials of ^[6] )

${\displaystyle E(e^{itX})=\left({\frac {a}{a-2it}}\right)^{\frac {p}{2}}{\frac {K_{p}\left({\sqrt {(a-2it)b}}\right)}{K_{p}\left({\sqrt {ab}}\right)}}}$

for ${\displaystyle t\in \mathbb {R} }$ where ${\displaystyle i}$ denotes the imaginary number.

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

${\displaystyle f(x;\mu ,\lambda )=\left[{\frac {\lambda }{2\pi x^{3}}}\right]^{1/2}\exp {\left({\frac {-\lambda (x-\mu )^{2}}{2\mu ^{2}x}}\right)}}$

is a GIG with ${\displaystyle a=\lambda /\mu ^{2}}$, ${\displaystyle b=\lambda }$, and ${\displaystyle p=-1/2}$. A Gamma distribution of the form

${\displaystyle g(x;\alpha ,\beta )=\beta ^{\alpha }{\frac {1}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}$

is a GIG with ${\displaystyle a=2\beta }$, ${\displaystyle b=0}$, and ${\displaystyle p=\alpha }$.

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 ${\displaystyle z}$, be GIG:

${\displaystyle P(z\mid a,b,p)=\operatorname {GIG} (z\mid a,b,p)}$

and let there be ${\displaystyle T}$ observed data points, ${\displaystyle X=x_{1},\ldots ,x_{T}}$, with normal likelihood function, conditioned on ${\displaystyle z:}$

${\displaystyle P(X\mid z,\alpha ,\beta )=\prod _{i=1}^{T}N(x_{i}\mid \alpha +\beta z,z)}$

where ${\displaystyle N(x\mid \mu ,v)}$ is the normal distribution, with mean ${\displaystyle \mu }$ and variance ${\displaystyle v}$. Then the posterior for ${\displaystyle z}$, given the data is also GIG:

${\displaystyle P(z\mid X,a,b,p,\alpha ,\beta )={\text{GIG}}\left(z\mid a+T\beta ^{2},b+S,p-{\frac {T}{2}}\right)}$

where ${\displaystyle \textstyle S=\sum _{i=1}^{T}(x_{i}-\alpha )^{2}}$.^{[note 1]}

Sichel distribution

The Sichel distribution results when the GIG is used as the mixing distribution for the Poisson parameter ${\displaystyle \lambda }$.^[10][11]

Notes

1. Due to the conjugacy, these details can be derived without solving integrals, by noting that

   ${\displaystyle P(z\mid X,a,b,p,\alpha ,\beta )\propto P(z\mid a,b,p)P(X\mid z,\alpha ,\beta )}$.

   Omitting all factors independent of ${\displaystyle z}$, 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 grandson of the mathematician Georges Henri Halphen.
4. Jørgensen, Bent (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics. Vol. 9. New York–Berlin: Springer-Verlag. ISBN 0-387-90665-7. MR 0648107.
5. Barndorff-Nielsen, O.; Halgreen, Christian (1977). "Infinite Divisibility of the Hyperbolic and Generalized Inverse Gaussian Distributions". Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete. 38: 309–311. doi:10.1007/BF00533162.
6. Pal, Subhadip; Gaskins, Jeremy (23 May 2022). "Modified Pólya-Gamma data augmentation for Bayesian analysis of directional data". Journal of Statistical Computation and Simulation. 92 (16): 3430–3451. doi:10.1080/00949655.2022.2067853. ISSN 0094-9655. S2CID 249022546.
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. Karlis, Dimitris (2002). "An EM type algorithm for maximum likelihood estimation of the normal–inverse Gaussian distribution". Statistics & Probability Letters. 57 (1): 43–52. doi:10.1016/S0167-7152(02)00040-8.
9. Barndorf-Nielsen, O. E. (1997). "Normal Inverse Gaussian Distributions and stochastic volatility modelling". Scand. J. Statist. 24 (1): 1–13. doi:10.1111/1467-9469.00045.
10. Sichel, Herbert S. (1975). "On a distribution law for word frequencies". Journal of the American Statistical Association. 70 (351a): 542–547. doi:10.1080/01621459.1975.10482469.
11. Stein, Gillian Z.; Zucchini, Walter; Juritz, June M. (1987). "Parameter estimation for the Sichel distribution and its multivariate extension". Journal of the American Statistical Association. 82 (399): 938–944. doi:10.1080/01621459.1987.10478520.

See also

• Inverse Gaussian distribution
• Gamma distribution