A Closed Form Approximation for Calculating the Percentage Points of the F and T Distributions

Paul D. Haines

Journal of the Royal Statistical Society Series C, 1988, vol. 37, issue 1, 95-100

Abstract: In defining the F distribution as a ratio of two independently distributed χ2 variates with m and n degrees of freedom for numerator and denominator respectively, the choice for approximating the distribution of χ2 is very important. Two approximations to this distribution are in common use, each of which selects an appropriate function of the variate which is normally distributed to a closer approximation than is χ2 itself. They are: (a) Fisher's result that √(2χ2) is approximately normally distributed with mean √(2v – 1) and unit variance; (b) Wilson and Hilferty's (1931) result that (χ2/v)1/3 is approximately normally distributed with mean 1 – 2/9v and variance 2/9v. We regard the second of these as being the more accurate approximation and which, by combining Hasting's (1955) fourth order approximation of the normal distribution of zero means and unit variance, permits us to construct a simple inversion process to acquire relatively accurate percentage points of the F distribution for given probability measures ranging from .90 to .99 for any selected numerator and denominator degrees of freedom.

Date: 1988
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