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

2002; Elsevier BV; Volume: 40; Issue: 1 Linguagem: Inglês

10.1016/s0167-9473(01)00097-4

1872-7352

Baibing Li, Elaine Martin,

Cyclone Separators and Fluid Dynamics

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(λkx;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 λ=λ(kx;n) is the shrinking factor. Numerical analysis indicates that for n/k⩾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 Scheffé–Tukey approximations. In addition, it is shown that the theoretical approximation error of the SFA, |F(x;k,n)−G(λkx;k)|, is O(1/n2) uniformly over x.