Generalized inverse Gaussian distribution

Generalized inverse Gaussian
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 %2B b/x)/2}
Mean \frac{\sqrt{b}\ K_{p%2B1}(\sqrt{a b}) }{ \sqrt{a}\ K_{p}(\sqrt{a b})}
Mode \frac{(p-1)%2B\sqrt{(p-1)^2%2Bab}}{a}
Variance \left(\frac{b}{a}\right)\left[\frac{K_{p%2B2}(\sqrt{ab})}{K_p(\sqrt{ab})}-\left(\frac{K_{p%2B1}(\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 %2B b/x)/2},\qquad x>0,

where Kp 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 Etienne Halphen.[1] It was rediscovered and popularised by Ole Barndorff-Nielsen, who called it the generalized inverse Gaussian distribution, and Herbert Sichel. It is also known as the Sichel distribution. Its statistical properties are discussed in Bent Jørgensen's lecture notes.[2]

The inverse Gaussian and gamma distributions are special cases of the generalized inverse Gaussian distribution for p = -1/2 and b = 0, respectively.

Entropy

The entropy of the generalized inverse Gaussian distribution is given as H(f(x))=\frac{1}{2} \log \left(\frac{b}{a}\right)%2B\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)}%2B\frac{\sqrt{a b}}{2 K_p\left(\sqrt{a b}\right)}\left( K_{p%2B1}\left(\sqrt{a b}\right) %2B 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

References

  1. ^ 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 
  2. ^ 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. MR0648107. 

See also
 
Benini · Benktander 1st kind · Benktander 2nd kind · Beta prime · Bose–Einstein · Burr · chi-squared · chi · Coxian · Dagum · Davis · Erlang · exponential · F · Fermi–Dirac · folded normal · Fréchet · Gamma · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-squared · hyper-exponential · hypoexponential · inverse chi-squared (scaled-inverse-chi-squared) · inverse Gaussian · inverse gamma · Kolmogorov · Lévy · log-Cauchy · log-Laplace · log-logistic · log-normal · Maxwell–Boltzmann · Maxwell speed · Mittag–Leffler · Nakagami · noncentral chi-squared · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda