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:

\begin{eqnarray*} g(x, \alpha, \beta, \delta, \mu) = \frac{\alpha\delta K_1 \left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}{\pi \sqrt{\delta^2 + (x - \mu)^2}} \, e^{\delta \sqrt{\alpha^2 - \beta^2} + \beta (x - \mu)} \end{eqnarray*}

In SciPy, this corresponds to a = \alpha \delta, b = \beta \delta, \text{loc} = \mu, \text{scale}=\delta.

Implementation: scipy.stats.norminvgauss