Applying series expansion to the inverse beta distribution to find percentiles of the F -distribution

Abstract

Let 0 ≤ 1 and F be the cumulative distribution function (cdf) of the F -Distribution. We wish to find x p such that F(x p |n 1 , n 2 ) = p , where n 1 and n 2 are the degrees of freedom. Traditionally, x p is found using a numerical root-finding method, such as Newton's method. In this paper, a procedure based on a series expansion for finding x p is given. The series expansion method has been applied to the normal, chi-square, and t distributions, but because of computational difficulties, it has not been applied to the F -Distribution. These problems have been overcome by making the standard transformation to the beta distribution. The procedure is explained in Sections 3 and 4. Empirical results of a comparison of CPU times are given in Section 5. The series expansion is compared to some of the standard root-finding methods. A table is given for p = .90.