Wilson-Hilferty transformation

For $X\sim {\chi }_{\nu }^2$, the Wilson-Hilferty transformation uses the approximationThe cube root reduces the right skewness of the chi-squared distribution. Inverting gives the approximate $p$-quantile $\nu [1-2/(9\nu )+z_p\sqrt{2/(9\nu )}{]}^3$, where $z_p$ is a standard normal quantile. This is an approximation, especially imperfect for extreme tails or small degrees of freedom. For a Gamma distribution with shape $\alpha$ and rate $\beta$, apply it to $X=2\beta Y$, which has $\nu =2\alpha$.^[ref]