Generalized inverse Gaussian distribution

In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function 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. 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. By setting θ = a b {displaystyle heta ={sqrt {ab}}} and η = b / a {displaystyle eta ={sqrt {b/a}}} , we can alternatively express the GIG distribution as where θ {displaystyle heta } is the concentration parameter while η {displaystyle eta } is the scaling parameter. Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible. The entropy of the generalized inverse Gaussian distribution is given as where [ d d ν K ν ( a b ) ] ν = p {displaystyle left_{ u =p}} is a derivative of the modified Bessel function of the second kind with respect to the order ν {displaystyle u } evaluated at ν = p {displaystyle u =p} 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 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