# Normal Inverse Gaussian Distribution#

The probability density function is given by:

\begin{eqnarray*} f(x; a, b) = \frac{a \exp\left(\sqrt{a^2 - b^2} + b x \right)}{\pi \sqrt{1 + x^2}} \, K_1\left(a * \sqrt{1 + x^2}\right), \end{eqnarray*}

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