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| {{Probability distribution |
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| name =Normal-inverse Gaussian (NIG)|
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| type =density|
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| pdf_image =|
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| cdf_image =|
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| parameters =<math>\mu</math> [[location parameter|location]] ([[real number|real]])<br/> <math>\alpha</math> tail heavyness (real)<br/> <math>\beta</math> asymmetry parameter (real)<br/> <math>\delta</math> [[scale parameter]] (real)<br/> <math>\gamma = \sqrt{\alpha^2 - \beta^2}</math>|
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| support =<math>x \in (-\infty; +\infty)\!</math>|
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| pdf =<math>\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)}</math> <br/><br/> <math>K_j</math> denotes a modified [[Bessel function]] of the second kind|
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| cdf =<!-- to do -->|
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| mean =<math>\mu + \delta \beta / \gamma</math>|
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| median =<!-- to do -->|
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| mode =|
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| variance =<math>\delta\alpha^2/\gamma^3</math>|
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| skewness =<math> 3 \beta /(\alpha \sqrt{\delta \gamma})</math>|
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| kurtosis =<math>3(1+4 \beta^2/\alpha^2)/(\delta\gamma)</math>|
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| entropy =<!-- to do -->|
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| mgf =<math>e^{\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +z)^2})}</math>|
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| char =<math>e^{i\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +iz)^2})}</math>|
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| }}
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| The '''normal-inverse Gaussian distribution (NIG)''' is [[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 introduced by [[Ole Barndorff-Nielsen]]<ref>{{cite journal|doi=10.1098/rspa.1977.0041|last=Barndorff-Nielsen|first=Ole|year=1977|title=Exponentially decreasing distributions for the logarithm of particle size|journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences|volume=353|issue=1674|pages=401–409|jstor=79167|publisher=The Royal Society}}</ref> and is a subclass of the [[generalised hyperbolic distribution]]. The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the [[normal distribution]], <math>N(\mu,\sigma^2),</math> arises as a special case by setting <math>\beta=0, \delta=\sigma^2\alpha,</math> and letting <math>\alpha\rightarrow\infty</math>.
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| The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. The class of normal-inverse Gaussian distributions is closed under convolution in the following sense. If <math>X_1</math> and <math>X_2</math> are [[statistical independence|independent]] [[random variable]]s that are NIG-distributed with the same values of the parameters <math>\alpha</math> and <math>\beta</math>, but possibly different values of the location and scale parameters, <math>\mu_1</math>, <math>\delta_1</math> and <math>\mu_2,</math> <math>\delta_2</math>, respectively, then <math>X_1 + X_2</math> is NIG-distributed with parameters <math>\alpha, </math> <math>\beta, </math><math>\mu_1+\mu_2</math> and <math>\delta_1 + \delta_2.</math>
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| 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]]), <math>W^{(\gamma)}(t)=W(t)+\gamma t</math>, we can define the inverse Gaussian process <math>A_t=\inf\{s>0:W^{(\gamma)}(s)=\delta t\}.</math> Then given a second independent drifting Brownian motion, <math>W^{(\beta)}(t)=\tilde W(t)+\beta t</math>, the normal-inverse Gaussian process is the time-changed process <math>X_t=W^{(\beta)}(A_t)</math>. The process <math>X(t)</math> 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]].
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| The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.
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| ==References==
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| <references/>
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| {{ProbDistributions|continuous-infinite}}
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| {{DEFAULTSORT:Normal-Inverse Gaussian Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Generalized hyperbolic distributions]]
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| [[Category:Probability distributions]]
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