Normal-inverse Gaussian distribution

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normal-inverse Gaussian (NIG)
Probability density function
Cumulative distribution function
Parameters	$μ$ location (real) $α$ (real) $β$ asymmetry parameter (real) $δ$ scale parameter (real) $\gamma = \sqrt{\alpha^2 - \beta^2}$
Support	$x \in (-\infty; +\infty)\!$
Probability density function (pdf)	$\frac{\alpha\delta K_1 \left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}{\pi \sqrt{\delta^2 + (x - \mu)^2}} \; e^{\delta \gamma + \beta (x - \mu)}$ $K λ$ denotes a modified Bessel function of the third kind
Cumulative distribution function (cdf)
Mean	$μ + δβ / γ$
Median
Mode
Variance	$δα 2 / γ 3$
Skewness	$3 \beta /(\alpha \sqrt{\delta \gamma})$
Excess kurtosis	$3(1 + 4β 2 / α 2) / (δγ)$
Entropy
Moment-generating function (mgf)	$e^{\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +z)^2})}$
Characteristic function

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 tails of the distribution decrease more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The normal-inverse Gaussian distributions form a subclass of the generalised hyperbolic distributions.

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 $X 1$ and $X 2$ are independent random variable 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 $δ 1 + δ 2 .$

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Categories: Continuous distributions | Generalized hyperbolic distributions

Normal-inverse Gaussian distribution

From Wikipedia, the free encyclopedia

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