Inverse Normal (Inverse Gaussian) Distribution

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

\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*}

\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 \(S\) and location parameter \(L\) by taking \(L=0\) and \(S\equiv\lambda,\) then \(\mu S\) is equal to \(\mu_{2}\) where \(\mu_{2}\) 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 \(\left(\sqrt{\beta_{1}}\right)\) and the kurtosis ( \(\beta_{2}-3\) ) are both functions only of \(\mu_{2}/\lambda=\mu S/S=\mu\) as shown here, while the variance and mean of the standard form here are transformed appropriately.

Implementation: scipy.stats.invgauss