Normal Inverse Gaussian Distribution#
The probability density function is given by:
where \(x\) is a real number, the parameter \(a\) is the tail heaviness and \(b\) is the asymmetry parameter satisfying \(a > 0\) and \(|b| \leq a\). \(K_1\) is the modified Bessel function of second kind (scipy.special.k1
).
A normal inverse Gaussian random variable with parameters \(a\) and \(b\) can be expressed as \(X = b V + \sqrt(V) X\) where \(X\) is norm(0,1) and \(V\) is invgauss(mu=1/sqrt(a**2 - b**2)). Hence, the normal inverse Gaussian distribution is a special case of normal variance-mean mixtures.
Another common parametrization of the distribution is given by the following expression of the pdf:
In SciPy, this corresponds to \(a = \alpha \delta, b = \beta \delta, \text{loc} = \mu, \text{scale}=\delta\).
Implementation: scipy.stats.norminvgauss