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Properties

Alternative parametrization

By setting θ = a b {\displaystyle \theta ={\sqrt {ab}}} and η = b / a {\displaystyle \eta ={\sqrt {b/a}}} , we can alternatively express the GIG distribution as

f ( x ) = 1 2 η K p ( θ ) ( x η ) p − 1 e − θ ( x / η + η / x ) / 2 , {\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 θ {\displaystyle \theta } is the concentration parameter while η {\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

H = 1 2 log ⁡ ( b a ) + log ⁡ ( 2 K p ( a b ) ) − ( p − 1 ) [ d d ν K ν ( a b ) ] ν = p K p ( a b ) + a b 2 K p ( a b ) ( K p + 1 ( a b ) + K p − 1 ( a b ) ) {\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 [ d d ν K ν ( a b ) ] ν = p {\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 ν {\displaystyle \nu } evaluated at ν = p {\displaystyle \nu =p}

Characteristic Function

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

E ( e i t X ) = ( a a − 2 i t ) p 2 K p ( ( a − 2 i t ) b ) K p ( a b ) {\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 t ∈ R {\displaystyle t\in \mathbb {R} } where i {\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

f ( x ; μ , λ ) = [ λ 2 π x 3 ] 1 / 2 exp ⁡ ( − λ ( x − μ ) 2 2 μ 2 x ) {\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 a = λ / μ 2 {\displaystyle a=\lambda /\mu ^{2}} , b = λ {\displaystyle b=\lambda } , and p = − 1 / 2 {\displaystyle p=-1/2} . A Gamma distribution of the form

g ( x ; α , β ) = β α 1 Γ ( α ) x α − 1 e − β x {\displaystyle g(x;\alpha ,\beta )=\beta ^{\alpha }{\frac {1}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}

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

Other special cases include the inverse-gamma distribution, for a = 0.⁸

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.⁹¹⁰ Let the prior distribution for some hidden variable, say z {\displaystyle z} , be GIG:

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

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

P ( X ∣ z , α , β ) = ∏ i = 1 T N ( x i ∣ α + β z , z ) {\displaystyle P(X\mid z,\alpha ,\beta )=\prod _{i=1}^{T}N(x_{i}\mid \alpha +\beta z,z)}

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

P ( z ∣ X , a , b , p , α , β ) = GIG ( z ∣ a + T β 2 , b + S , p − T 2 ) {\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 S = ∑ i = 1 T ( x i − α ) 2 {\displaystyle \textstyle S=\sum _{i=1}^{T}(x_{i}-\alpha )^{2}} .¹¹

Sichel distribution

The Sichel distribution results when the GIG is used as the mixing distribution for the Poisson parameter λ {\displaystyle \lambda } .¹²¹³

Notes

See also

• Inverse Gaussian distribution
• Gamma distribution

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). /wiki/Doi_(identifier) ↩

3. Étienne Halphen was the grandson of the mathematician Georges Henri Halphen. /wiki/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. 0-387-90665-7 ↩

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. /wiki/Doi_(identifier) ↩

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. https://www.tandfonline.com/doi/abs/10.1080/00949655.2022.2067853?journalCode=gscs20 ↩

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 978-0-471-58495-7 ↩

8. 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 978-0-471-58495-7 ↩

9. 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. /wiki/Doi_(identifier) ↩

10. 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. /wiki/Doi_(identifier) ↩

11. Due to the conjugacy, these details can be derived without solving integrals, by noting that P ( z ∣ X , a , b , p , α , β ) ∝ P ( z ∣ a , b , p ) P ( X ∣ z , α , β ) {\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 z {\displaystyle z} , the right-hand-side can be simplified to give an un-normalized GIG distribution, from which the posterior parameters can be identified. ↩

12. 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. /wiki/Doi_(identifier) ↩

13. 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. /wiki/Doi_(identifier) ↩