An asymptotic expansion for the upper percentage points of the $\chi ^{2}$-distribution

Abstract:

An asymptotic development is given for estimating the value of the variable $\chi$ for which the ${\chi ^2}$-distribution \[ Q({\chi ^2},v) = \frac {1}{{\Gamma (v/2)}}\int _{{\chi ^2}/2}^\infty {t^{v/2 - 1}}{e^{ - t}}dt\] assumes a preassigned value $\alpha$, in the region where the quantity $\eta = - \ln [\Gamma (v/2)\alpha ]$ satisfies \[ \eta > > \ln \eta .\] This development generalizes a similar one given by Blair and coauthors [2] for the case $v = 1$. It is also shown how the estimates thus obtained may be used in conjunction with various iterative schemes to give more accurate values.