Approximating the F distribution via a general version of the modified signed log-likelihood ratio statistic

A simple normal approximation for the cumulative distribution function of the F distribution is obtained via a general version of the modified signed log-likelihood ratio statistic. This approximation exhibits remarkable accuracy even when the degrees of freedom are small. Using the same methodology, but with a simpler set up, simple and accurate normal approximations to the cumulative distribution functions of the Student t and [chi]2 distributions can also be obtained.