Generalized inverse Gaussian distribution

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

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

f(x) = \frac{(a/b)^{p/2}}{2 K_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. It is also known as the Sichel distribution, after Herbert Sichel.^[4] Its statistical properties are discussed in Bent Jørgensen's lecture notes.^[5]

Properties

Summation

Barndorff-Nielsen and Halgreen proved that the GIG distribution has Infinite divisibility^[6]

Entropy

The entropy of the generalized inverse Gaussian distribution is given as

H(f(x))=\frac{1}{2} \log \left(\frac{b}{a}\right)+\log \left(2 K_p\left(\sqrt{a b}\right)\right)- (p-1) \frac{\left[\frac{d}{d\nu}K_\nu\left(\sqrt{ab}\right)\right]_{\nu=p}}{K_p\left(\sqrt{a b}\right)}+\frac{\sqrt{a b}}{2 K_p\left(\sqrt{a b}\right)}\left( K_{p+1}\left(\sqrt{a b}\right) + K_{p-1}\left(\sqrt{a b}\right)\right)

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

Differential equation

The pdf of the generalized inverse Gaussian distribution is a solution to the following differential equation:

\left\{{\begin{array}{l}f(x)(x(ax-2p+2)-b)+2x^{2}f'(x)=0,\\f(1)={\frac {e^{{{\frac {1}{2}}(-a-b)}}\left({\frac {a}{b}}\right)^{{p/2}}}{2K_{p}\left({\sqrt {ab}}\right)}}\end{array}}\right\}

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

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

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

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

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

Other special cases include the inverse-gamma distribution, for a=0, and the hyperbolic distribution, for p=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 $z$ , be GIG:

P(z|a,b,p) = \text{GIG}(z|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|z,\alpha,\beta) = \prod_{i=1}^T N(x_i|\alpha+\beta z,z)

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

P(z|X,a,b,p,\alpha ,\beta )={\text{GIG}}(z|a+T\beta ^{2},b+S,p-{\tfrac {T}{2}})

where $\textstyle S = \sum_{i=1}^T (x_i-\alpha)^2$ .^{[note 1]}

Notes

↑ Due to the conjugacy, these details can be derived without solving integrals, by noting that
$P(z|X,a,b,p,\alpha,\beta)\propto P(z|a,b,p)P(X|z,\alpha,\beta)$ .
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.

References

↑ Seshadri, V. (1997). "Halphen's laws". In Kotz, S.; Read, C. B.; Banks, D. L. Encyclopedia of Statistical Sciences, Update Volume 1. New York: Wiley. pp. 302–306.
↑ 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).
↑ Étienne Halphen was the uncle of the mathematician Georges Henri Halphen.
↑ Sichel, H.S., Statistical valuation of diamondiferous deposits, Journal of the South African Institute of Mining and Metallurgy 1973
↑ Jørgensen, Bent (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics. 9. New York–Berlin: Springer-Verlag. ISBN 0-387-90665-7. MR 0648107.
↑ O. Barndorff-Nielsen and Christian Halgreen, Infinite Divisibility of the Hyperbolic and Generalized Inverse Gaussian Distributions, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 1977
1 2 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
↑ Dimitris Karlis, "An EM type algorithm for maximum likelihood estimation of the normal–inverse Gaussian distribution", Statistics & Probability Letters 57 (2002) 43–52.
↑ Barndorf-Nielsen, O.E., 1997. Normal Inverse Gaussian Distributions and stochastic volatility modelling. Scand. J. Statist. 24, 1–13.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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