Normal-inverse Gaussian distribution

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.^[2] In the next year Barndorff-Nielsen published the NIG in another paper.^[3] It was introduced in the mathematical finance literature in 1997.^[4]

The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.^[5]

The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.^{[6] [7]}

This class is closed under affine transformations, since it is a particular case of the Generalized hyperbolic distribution, which has the same property. If

${\displaystyle x\sim {\mathcal {NIG}}(\alpha ,\beta ,\delta ,\mu ){\text{ and }}y=ax+b,}${\displaystyle x\sim {\mathcal {NIG}}(\alpha ,\beta ,\delta ,\mu ){\text{ and }}y=ax+b,}

then^[8]

${\displaystyle y\sim {\mathcal {NIG}}{\bigl (}{\frac {\alpha }{\left|a\right|}},{\frac {\beta }{a}},\left|a\right|\delta ,a\mu +b{\bigr )}.}${\displaystyle y\sim {\mathcal {NIG}}{\bigl (}{\frac {\alpha }{\left|a\right|}},{\frac {\beta }{a}},\left|a\right|\delta ,a\mu +b{\bigr )}.}

This class is infinitely divisible, since it is a particular case of the Generalized hyperbolic distribution, which has the same property.

The class of normal-inverse Gaussian distributions is closed under convolution in the following sense:^[9] if ${\displaystyle X_{1}}${\displaystyle X_{1}} and ${\displaystyle X_{2}}${\displaystyle X_{2}} are independent random variables that are NIG-distributed with the same values of the parameters ${\displaystyle \alpha }${\displaystyle \alpha } and ${\displaystyle \beta }${\displaystyle \beta }, but possibly different values of the location and scale parameters, ${\displaystyle \mu _{1}}${\displaystyle \mu _{1}}, ${\displaystyle \delta _{1}}${\displaystyle \delta _{1}} and ${\displaystyle \mu _{2},}${\displaystyle \mu _{2},} ${\displaystyle \delta _{2}}${\displaystyle \delta _{2}}, respectively, then ${\displaystyle X_{1}+X_{2}}${\displaystyle X_{1}+X_{2}} is NIG-distributed with parameters ${\displaystyle \alpha ,}${\displaystyle \alpha ,} ${\displaystyle \beta ,}${\displaystyle \beta ,}${\displaystyle \mu _{1}+\mu _{2}}${\displaystyle \mu _{1}+\mu _{2}} and ${\displaystyle \delta _{1}+\delta _{2}.}${\displaystyle \delta _{1}+\delta _{2}.}

The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the normal distribution, ${\displaystyle N(\mu ,\sigma ^{2}),}${\displaystyle N(\mu ,\sigma ^{2}),} arises as a special case by setting ${\displaystyle \beta =0,\delta =\sigma ^{2}\alpha ,}${\displaystyle \beta =0,\delta =\sigma ^{2}\alpha ,} and letting ${\displaystyle \alpha \rightarrow \infty }${\displaystyle \alpha \rightarrow \infty }.

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), ${\displaystyle W^{(\gamma )}(t)=W(t)+\gamma t}${\displaystyle W^{(\gamma )}(t)=W(t)+\gamma t}, we can define the inverse Gaussian process ${\displaystyle A_{t}=\inf\{s>0:W^{(\gamma )}(s)=\delta t\}.}${\displaystyle A_{t}=\inf\{s>0:W^{(\gamma )}(s)=\delta t\}.} Then given a second independent drifting Brownian motion, ${\displaystyle W^{(\beta )}(t)={\tilde {W}}(t)+\beta t}${\displaystyle W^{(\beta )}(t)={\tilde {W}}(t)+\beta t}, the normal-inverse Gaussian process is the time-changed process ${\displaystyle X_{t}=W^{(\beta )}(A_{t})}${\displaystyle X_{t}=W^{(\beta )}(A_{t})}. The process ${\displaystyle X(t)}${\displaystyle X(t)} at time ${\displaystyle t=1}${\displaystyle t=1} has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of Lévy processes.

Let ${\displaystyle {\mathcal {IG}}}${\displaystyle {\mathcal {IG}}} denote the inverse Gaussian distribution and ${\displaystyle {\mathcal {N}}}${\displaystyle {\mathcal {N}}} denote the normal distribution. Let ${\displaystyle z\sim {\mathcal {IG}}(\delta ,\gamma )}${\displaystyle z\sim {\mathcal {IG}}(\delta ,\gamma )}, where ${\displaystyle \gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}}${\displaystyle \gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}}; and let ${\displaystyle x\sim {\mathcal {N}}(\mu +\beta z,z)}${\displaystyle x\sim {\mathcal {N}}(\mu +\beta z,z)}, then ${\displaystyle x}${\displaystyle x} follows the NIG distribution, with parameters, ${\displaystyle \alpha ,\beta ,\delta ,\mu }${\displaystyle \alpha ,\beta ,\delta ,\mu }. This can be used to generate NIG variates by ancestral sampling. It can also be used to derive an EM algorithm for maximum-likelihood estimation of the NIG parameters.^[10]

• Continuous distributions

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