An approximation to the F distribution using the chi-square distribution

For the cumulative distribution function (c.d.f.) of the F distribution, F(x; k, n), with associated degrees of freedom, k and n, a shrinking factor approximation (SFA), G(lambdakx; k), is proposed for large n and any fixed k, where G(x; k) is the chi-square c.d.f. with degrees of freedom, k, and lambda=lambda(kx; n) is the shrinking factor. Numerical analysis indicates that for n/k greater than or equal to 3, approximation accuracy of the SFA is to the fourth decimal place for most small values of k. This is a substantial improvement on the accuracy that is achievable using the normal, ordinary chi-square, and Scheffe-Tukey approximations. In addition, it is shown that the theoretical approximation error of the SFA, \f(x;k,n)-G(lambdakx;k)\, is O(1/n(2)) uniformly over x. (C) 2002 Elsevier Science B.V. All rights reserved.