Parameters a > 0, b > 0, p real | Support x > 0 | |
![]() | ||
PDF f ( x ) = ( a / b ) p / 2 2 K p ( a b ) x ( p − 1 ) e − ( a x + b / x ) / 2 {displaystyle f(x)={rac {(a/b)^{p/2}}{2K_{p}({sqrt {ab}})}}x^{(p-1)}e^{-(ax+b/x)/2}} Mean E [ x ] = b K p + 1 ( a b ) a K p ( a b ) {displaystyle operatorname {E} [x]={rac {{sqrt {b}} K_{p+1}({sqrt {ab}})}{{sqrt {a}} K_{p}({sqrt {ab}})}}} E [ x − 1 ] = a K p + 1 ( a b ) b K p ( a b ) − 2 p b {displaystyle operatorname {E} [x^{-1}]={rac {{sqrt {a}} K_{p+1}({sqrt {ab}})}{{sqrt {b}} K_{p}({sqrt {ab}})}}-{rac {2p}{b}}} E [ ln x ] = ln b a + ∂ ∂ p ln K p ( a b ) {displaystyle operatorname {E} [ln x]=ln {rac {sqrt {b}}{sqrt {a}}}+{rac {partial }{partial p}}ln K_{p}({sqrt {ab}})} Mode ( p − 1 ) + ( p − 1 ) 2 + a b a {displaystyle {rac {(p-1)+{sqrt {(p-1)^{2}+ab}}}{a}}} Variance ( b a ) [ K p + 2 ( a b ) K p ( a b ) − ( K p + 1 ( a b ) K p ( a b ) ) 2 ] {displaystyle left({rac {b}{a}}ight)left[{rac {K_{p+2}({sqrt {ab}})}{K_{p}({sqrt {ab}})}}-left({rac {K_{p+1}({sqrt {ab}})}{K_{p}({sqrt {ab}})}}ight)^{2}ight]} |
In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function
Contents
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. 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. Its statistical properties are discussed in Bent Jørgensen's lecture notes.
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
Differential equation
The pdf of the generalized inverse Gaussian distribution is a solution to the following differential equation:
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
is a GIG with
Other special cases include the inverse-gamma distribution, for a = 0, and the hyperbolic distribution, for p = 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
and let there be
where
where