The deviate,
${f}_{p}$, associated with the lower tail probability,
$p$, of the
$F$-distribution with degrees of freedom
${\nu}_{1}$ and
${\nu}_{2}$ is defined as the solution to
where
${\nu}_{1},{\nu}_{2}>0$;
$0\le {f}_{p}<\infty $.
The value of
${f}_{p}$ is computed by means of a transformation to a beta distribution,
${P}_{\beta}\left(B\le \beta :a,b\right)$:
and using a call to
nag_deviates_beta (g01fec).
For very large values of both
${\nu}_{1}$ and
${\nu}_{2}$, greater than
${10}^{5}$, a normal approximation is used. If only one of
${\nu}_{1}$ or
${\nu}_{2}$ is greater than
${10}^{5}$ then a
${\chi}^{2}$ approximation is used; see
Abramowitz and Stegun (1972).
The result should be accurate to five significant digits.
For higher accuracy
nag_deviates_beta (g01fec) can be used along with the transformations given in
Section 3.