.. _continuous-norminvgauss:

Normal Inverse Gaussian Distribution

The probability density function is given by:

.. math:: :nowrap:

\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 :math:x is a real number, the parameter :math:a is the tail heaviness and :math:b is the asymmetry parameter satisfying :math:a > 0 and :math:|b| \leq a. :math:K_1 is the modified Bessel function of second kind (scipy.special.k1).

A normal inverse Gaussian random variable with parameters :math:a and :math:b can be expressed as :math:X = b V + \sqrt(V) X where :math:X is norm(0,1) and :math: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:

.. math:: :nowrap:

\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 :math:a = \alpha \delta, b = \beta \delta, \text{loc} = \mu, \text{scale}=\delta.

Implementation: scipy.stats.norminvgauss