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
	2K_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 as

f(x)=

	1	\left(
	2ηK_p(\theta)

	x
	η

\right)^p-1e^-\theta(x/η,

where

\theta

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

\begin{align} H=

	1
	2

log\left(

	b
	a

\right)&{}+log\left(2K_{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\simGIG(p,a,b)

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

E(e^itX)=\left(

	a
	a-2it

	p
	2

\right)

	K_p\left(\sqrt{(a-2it)b
	\right)}{

K_p\left(\sqrt{ab}\right)}

for

t\inR

where

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\pix³

\right]^1/2\exp{\left(

	-λ(x-\mu)²
	2\mu²x

\right)}

is a GIG with

a=λ/\mu²

b=λ

, and

p=-1/2

. A Gamma distribution of the form

g(x;\alpha,\beta)=\beta^\alpha

	1
	\Gamma(\alpha)

x^\alpha-1e^-\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

, be GIG:

P(z\mida,b,p)=\operatorname{GIG}(z\mida,b,p)

and let there be

observed data points,

X=x_1,\ldots,x_T

, with normal likelihood function, conditioned on

P(X\midz,\alpha,\beta)=

	T
\prod
	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

. Then the posterior for

, given the data is also GIG:

P(z\midX,a,b,p,\alpha,\beta)=GIG\left(z\mid

2,b+S,p-	T
	2

a+T\beta

\right)

where

styleS=

	T
\sum
	i=1

	2
(x
	i-\alpha)

.^[9]

Sichel distribution

The Sichel distribution results when the GIG is used as the mixing distribution for the Poisson parameter

.^[10] ^[11]

Notes and References

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 .
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 .
Étienne Halphen was the grandson of the mathematician Georges Henri Halphen.
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.
O. . Barndorff-Nielsen . Christian . Halgreen . Infinite Divisibility of the Hyperbolic and Generalized Inverse Gaussian Distributions . Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete . 1977 . 38 . 309–311 . 10.1007/BF00533162 .
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.
Dimitris . Karlis . An EM type algorithm for maximum likelihood estimation of the normal–inverse Gaussian distribution . Statistics & Probability Letters . 57 . 1 . 2002 . 43–52 . 10.1016/S0167-7152(02)00040-8 .
Barndorf-Nielsen . O. E. . 1997 . Normal Inverse Gaussian Distributions and stochastic volatility modelling . Scand. J. Statist. . 24 . 1 . 1–13 . 10.1111/1467-9469.00045 .
Due to the conjugacy, these details can be derived without solving integrals, by noting that
P(z\midX,a,b,p,\alpha,\beta)\proptoP(z\mida,b,p)P(X\midz,\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.
Sichel . Herbert S. . 1975 . On a distribution law for word frequencies . Journal of the American Statistical Association . 70 . 351a . 542-547 . 10.1080/01621459.1975.10482469 .
Stein . Gillian Z. . Walter . Zucchini . June M. . Juritz . 1987 . Parameter estimation for the Sichel distribution and its multivariate extension . Journal of the American Statistical Association . 82 . 399 . 938-944 . 10.1080/01621459.1987.10478520 .

Generalized inverse Gaussian distribution explained

Properties

Alternative parametrization

Summation

Entropy

Characteristic Function

Related distributions

Special cases

Conjugate prior for Gaussian

Sichel distribution

See also

Notes and References