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

Let <italic>0 ≤</italic> 1 and <italic>F</italic> be the cumulative distribution function (cdf) of the <italic>F</italic>-Distribution. We wish to find <italic>x<subscrpt>p</subscrpt></italic> such that <italic>F(x<subscrpt>p</subscrpt>|n<subscrpt>1</subscrpt>, n<subscrpt>2</subscrpt>) = p</italic>, where <italic>n<subscrpt>1</subscrpt></italic> and <italic>n<subscrpt>2</subscrpt></italic> are the degrees of freedom. Traditionally, <italic>x<subscrpt>p</subscrpt></italic> 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 <italic>x<subscrpt>p</subscrpt></italic>is given. The series expansion method has been applied to the normal, chi-square, and <italic>t</italic> distributions, but because of computational difficulties, it has not been applied to the <italic>F</italic>-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 <italic>p</italic> = .90.