Parameters μ
{\displaystyle \mu }
location (real)
α
{\displaystyle \alpha }
tail heaviness (real)
β
{\displaystyle \beta }
asymmetry parameter (real)
δ
{\displaystyle \delta }
scale parameter (real)
γ
=
α
2
−
β
2
{\displaystyle \gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}} Support x
∈
(
−
∞
;
+
∞
)
{\displaystyle x\in (-\infty ;+\infty )\!} PDF α
δ
K
1
(
α
δ
2
+
(
x
−
μ
)
2
)
π
δ
2
+
(
x
−
μ
)
2
e
δ
γ
+
β
(
x
−
μ
)
{\displaystyle {\frac {\alpha \delta K_{1}\left(\alpha {\sqrt {\delta ^{2}+(x-\mu )^{2}}}\right)}{\pi {\sqrt {\delta ^{2}+(x-\mu )^{2}}}}}\;e^{\delta \gamma +\beta (x-\mu )}}
K
j
{\displaystyle K_{j}}
denotes a modified Bessel function of the third kind Mean μ
+
δ
β
/
γ
{\displaystyle \mu +\delta \beta /\gamma } Variance δ
α
2
/
γ
3
{\displaystyle \delta \alpha ^{2}/\gamma ^{3}} Skewness 3
β
/
(
α
δ
γ
)
{\displaystyle 3\beta /(\alpha {\sqrt {\delta \gamma }})} | ||
The normal-inverse Gaussian distribution (NIG) 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.
Contents
The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.
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.
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
Related Distributions
The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the normal distribution,
Stochastic Process
The normal-inverse Gaussian distribution can also be seen as the marginal distribution of the normal-inverse Gaussian process which provides an alternative way of explicitly constructing it. Starting with a drifting Brownian motion (Wiener process),