Generalized inverse Gaussian distribution explained

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

f(x) = ((a/b) ^{p/2})/(2 K _{p(\sqrt{ab}) )} x ^{(p-1)} e ^{-(ax} , 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

\theta = \sqrt{ab} and η = \sqrt{b/a} , we can alternatively express the GIG distribution asf(x) = (1)/(\left( 2η K _{p(\theta)}) (x)/(η) \right) ^{p-1} e ^{-\theta(x/η} ,

where

\theta is the concentration parameter while η 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

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

where

\left[ (d)/(d\nu) K _{\nu\left(\sqrt{a} b}\right)\right] _{\nu=p} is a derivative of the modified Bessel function of the second kind with respect to the order \nu evaluated at \nu = p

Characteristic Function

The characteristic of a random variable

X\sim GIG(p, a, b) is given as(for a derivation of the characteristic function, see supplementary materials of ^[6]) E(e ^{itX} ) = \left( (a)/(a -2 it) \right) ^{(p)/(2)} (K _{p} \left( \sqrt{(a -2 it)b)/(\right)}{) K _{p} \left( \sqrt{ab} \right) }

for

t \in R where 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. Specifically, an inverse Gaussian distribution of the form

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

is a GIG with

a = λ /\mu ^{2} , b = λ , and p =-1 / 2 . A Gamma distribution of the formg(x;\alpha,\beta) = \beta ^{\alpha} (1)/(\Gamma(\alpha)) x ^{\alpha-1} e ^{-\beta} is a GIG with a = 2 \beta , b = 0 , and p = \alpha .

Other special cases include the inverse-gamma distribution, for a = 0.

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.^{[7] [8]} Let the prior distribution for some hidden variable, say

z , be GIG:P(z\mid a,b,p) = \operatorname{GIG}(z\mid a,b,p) and let there be T observed data points, X = x _{1,\ldots,x} _{T} , with normal likelihood function, conditioned on z: P(X\mid z,\alpha,\beta) = \prod ^{T} _{i =1} N(x _{i\mid\alpha+\beta} z,z)

where

N(x\mid\mu,v) is the normal distribution, with mean \mu and variance v . Then the posterior for z , given the data is also GIG:P(z\mid X,a,b,p,\alpha,\beta) = GIG \left(z\mid a + T\beta ^{2 ,b + S,p - (T)/(2)} \right) where style S = \sum ^{T} _{i =1} (x ^{2} _{i - \alpha)} .^[9]

Sichel distribution

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

λ .

See also

• Inverse Gaussian distribution
• Gamma distribution

Notes and References

1. Book: Seshadri , V. . Halphen's laws . Kotz . S. . Read . C. B. . Banks . D. L. . Encyclopedia of Statistical Sciences, Update Volume 1 . 302–306 . Wiley . New York . 1997 .
2. Perreault . L. . Bobée . B. . Rasmussen . P. F. . 10.1061/(ASCE)1084-0699(1999)4:3(189) . Halphen Distribution System. I: Mathematical and Statistical Properties . Journal of Hydrologic Engineering . 4 . 3 . 189 . 1999 .
3. Étienne Halphen was the grandson of the mathematician Georges Henri Halphen.
4. Book: Jørgensen , Bent . Statistical Properties of the Generalized Inverse Gaussian Distribution . Springer-Verlag . 1982 . New York–Berlin . Lecture Notes in Statistics . 9 . 0-387-90665-7 . 0648107.
5. O. Barndorff-Nielsen and Christian Halgreen, Infinite Divisibility of the Hyperbolic and Generalized Inverse Gaussian Distributions, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 1977
6. Pal . Subhadip . Gaskins . Jeremy . Modified Pólya-Gamma data augmentation for Bayesian analysis of directional data . Journal of Statistical Computation and Simulation . 23 May 2022 . 92 . 16 . 3430–3451 . 10.1080/00949655.2022.2067853 . 249022546 . 0094-9655.
7. Dimitris Karlis, "An EM type algorithm for maximum likelihood estimation of the normal–inverse Gaussian distribution", Statistics & Probability Letters 57 (2002) 43–52.
8. Barndorf-Nielsen, O.E., 1997. Normal Inverse Gaussian Distributions and stochastic volatility modelling. Scand. J. Statist. 24, 1–13.
9. Due to the conjugacy, these details can be derived without solving integrals, by noting that 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 z , the right-hand-side can be simplified to give an un-normalized GIG distribution, from which the posterior parameters can be identified.
10. Sichel, Herbert S, 1975. "On a distribution law for word frequencies." Journal of the American Statistical Association 70.351a: 542-547.
11. Stein, Gillian Z., Walter Zucchini, and June M. Juritz, 1987. "Parameter estimation for the Sichel distribution and its multivariate extension." Journal of the American Statistical Association 82.399: 938-944.