.. _continuous-invgauss:

Inverse Normal (Inverse Gaussian) Distribution

The standard form involves the shape parameter :math:\mu (in most definitions, :math:L=0.0 is used). (In terms of the regress documentation :math:\mu=A/B ) and :math:B=S and :math:L is not a parameter in that distribution. A standard form is :math:x>0

.. math:: :nowrap:

\begin{eqnarray*} f\left(x;\mu\right) & = & \frac{1}{\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-\mu\right)^{2}}{2x\mu^{2}}\right).\\
F\left(x;\mu\right) & = & \Phi\left(\frac{1}{\sqrt{x}}\frac{x-\mu}{\mu}\right)+\exp\left(\frac{2}{\mu}\right)\Phi\left(-\frac{1}{\sqrt{x}}\frac{x+\mu}{\mu}\right)\\
G\left(q;\mu\right) & = & F^{-1}\left(q;\mu\right)\end{eqnarray*}

.. math:: :nowrap:

\begin{eqnarray*} \mu & = & \mu\\
\mu_{2} & = & \mu^{3}\\
\gamma_{1} & = & 3\sqrt{\mu}\\
\gamma_{2} & = & 15\mu\\
m_{d} & = & \frac{\mu}{2}\left(\sqrt{9\mu^{2}+4}-3\mu\right)\end{eqnarray*}

This is related to the canonical form or JKB "two-parameter" inverse Gaussian when written in it's full form with scale parameter :math:S and location parameter :math:L by taking :math:L=0 and :math:S\equiv\lambda, then :math:\mu S is equal to :math:\mu_{2} where :math:\mu_{2} is the parameter used by JKB. We prefer this form because of it's consistent use of the scale parameter. Notice that in JKB the skew :math:\left(\sqrt{\beta_{1}}\right) and the kurtosis ( :math:\beta_{2}-3 ) are both functions only of :math:\mu_{2}/\lambda=\mu S/S=\mu as shown here, while the variance and mean of the standard form here are transformed appropriately.

Implementation: scipy.stats.invgauss