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 xp such that F(xp|n1, n2) = p, where n1 and n2 are the degrees of freedom. Traditionally, xp 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 xpis 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.

REFERENCES

• 1: CARTER, A.H. 1947. Approximations to percentage points of the z-distribution. B~ometrika 34, 2 tDec ), 352-358
• 2: GOLDBERG, H., AND LEVINE, H. 1946. Approximate formulas for the percentage points and normalization of t and ~2. An. Math. Star. 17, 4 (Dec), 571-572.
• 3: HILL, G. W., AND D^VIS, A.W. 1968. Generahzed asymptotic expansions of a Cornish-Fisher type An Moth Stat. 39, 8 (Aug.), 1264-1273.
• 4: MAJUMDER, J. L., AND BATTACHAILIEE, G. P 1973 Algorithm AS64. Inverse of the incomplete beta function ratio. Appl. S'tat. 22, 3, 411 414.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

Additional Classification:
  G. Mathematics of Computing

General Terms:
Algorithms, Performance

Keywords:
F-distribution, Newton's method, Taylor series, cumulative distribution function, distribution function, root finding methods