Normal inverse Gaussian distribution - Alchetron, the free social encyclopedia


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.

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 X 1 and X 2 are independent random variables that are NIG-distributed with the same values of the parameters α and β , but possibly different values of the location and scale parameters, μ 1 , δ 1 and μ 2 , δ 2 , respectively, then X 1 + X 2 is NIG-distributed with parameters α , β , μ 1 + μ 2 and δ 1 + δ 2 .

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 ) , arises as a special case by setting β = 0 , δ = σ 2 α , and letting α → ∞ .

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), W ( γ ) ( t ) = W ( t ) + γ t , we can define the inverse Gaussian process A t = inf { s > 0 : W ( γ ) ( s ) = δ t } . Then given a second independent drifting Brownian motion, W ( β ) ( t ) = W ~ ( t ) + β t , the normal-inverse Gaussian process is the time-changed process X t = W ( β ) ( A t ) . The process X ( t ) at time 1 has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of Lévy processes.

References

Normal-inverse Gaussian distribution Wikipedia

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