In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function
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 Étienne Halphen.Étienne Halphen was the grandson of the mathematician Georges Henri Halphen. 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.
Properties Alternative parametrization By setting and , we can alternatively express the GIG distribution as
where is the concentration parameter while is the scaling parameter.
Summation Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible.
Entropy The entropy of the generalized inverse Gaussian distribution is given as
where is a derivative of the modified Bessel function of the second kind with respect to the order evaluated at
Characteristic Function The characteristic of a random variable is given as
for where denotes the imaginary unit.
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
is a GIG with , , and . A gamma distribution of the form
is a GIG with , , and .
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. Let the prior distribution for some hidden variable, say , be GIG:
and let there be observed data points, , with normal likelihood function, conditioned on
where is the normal distribution, with mean and variance . Then the posterior for , given the data is also GIG:
where .Due to the conjugacy, these details can be derived without solving integrals, by noting that . Omitting all factors independent of , the right-hand-side can be simplified to give an un-normalized GIG distribution, from which the posterior parameters can be identified.
Sichel distribution The Sichel distribution results when the GIG is used as the mixing distribution for the Poisson parameter .
Notes
References
See also Inverse Gaussian distribution Gamma distribution