Generalized inverse Gaussian distribution

Probability distribution
name =Generalized inverse Gaussian
type =density
pdf_

cdf_

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}
cdf =
mean =$frac\{sqrt\{b\}\; K\_\{-1-p\}(sqrt\{a\; b\})\; \}\{\; sqrt\{a\}\; K\_\{p\}(sqrt\{a\; b\})\}$
median =
mode =
variance =
skewness =
kurtosis =
entropy =
mgf =
char =
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 third 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 [Citation
last = Seshadri
first = V.
contribution = Halphen's laws
editor-last = Kotz
editor-first = S.
editor2-last = Read
editor2-first = C. B.
editor3-last = Banks
editor3-first = D. L.
title = Encyclopedia of Statistical Sciences, Update Volume 1
pages = 302–306
publisher = Wiley
place = New York
year = 1997] . 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 [cite book
last = Jørgensen
first = Bent
title = Statistical Properties of the Generalized Inverse Gaussian Distribution
publisher = Springer-Verlag
date = 1982
location = New York–Berlin
series = Lecture Notes in Statistics
volume = 9
isbn = 0-387-90665-7
id = MathSciNet | id = 0648107] .

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