The normal-inverse Gaussian distribution (NIG, also known as the normal-Wald distribution) is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. The NIG distribution was noted by Blaesild in 1977 as a subclass of the generalised hyperbolic distribution discovered by Ole Barndorff-Nielsen. In the next year Barndorff-Nielsen published the NIG in another paper. It was introduced in the mathematical finance literature in 1997. The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.
Properties
Moments The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.
Linear transformation This class is closed under affine transformations, since it is a particular case of the Generalized hyperbolic distribution, which has the same property. If
x ∼ N I G ( α , β , δ , μ ) and y = a x + b , {\displaystyle x\sim {\mathcal {NIG}}(\alpha ,\beta ,\delta ,\mu ){\text{ and }}y=ax+b,} then
y ∼ N I G ( α | a | , β a , | a | δ , a μ + b ) . {\displaystyle y\sim {\mathcal {NIG}}{\bigl (}{\frac {\alpha }{\left|a\right|}},{\frac {\beta }{a}},\left|a\right|\delta ,a\mu +b{\bigr )}.}
Summation This class is infinitely divisible, since it is a particular case of the Generalized hyperbolic distribution, which has the same property.
Convolution The class of normal-inverse Gaussian distributions is closed under convolution in the following sense: if X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} are independent random variables that are NIG-distributed with the same values of the parameters α {\displaystyle \alpha } and β {\displaystyle \beta } , but possibly different values of the location and scale parameters, μ 1 {\displaystyle \mu _{1}} , δ 1 {\displaystyle \delta _{1}} and μ 2 , {\displaystyle \mu _{2},} δ 2 {\displaystyle \delta _{2}} , respectively, then X 1 + X 2 {\displaystyle X_{1}+X_{2}} is NIG-distributed with parameters α {\displaystyle \alpha } , β {\displaystyle \beta } , μ 1 + μ 2 {\displaystyle \mu _{1}+\mu _{2}} and δ 1 + δ 2 . {\displaystyle \delta _{1}+\delta _{2}.}
Related distributions The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the normal distribution, N ( μ , σ 2 ) , {\displaystyle N(\mu ,\sigma ^{2}),} arises as a special case by setting β = 0 , δ = σ 2 α , {\displaystyle \beta =0,\delta =\sigma ^{2}\alpha ,} and letting α → ∞ {\displaystyle \alpha \rightarrow \infty } .
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