Approximations for the doubly noncentral-F distribution

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Abstract
The approximate normality of the cube root of the noncentral chi-square observed by Aty (1954) and an Edgeworth-series expansion are used to derive an approximation for the doubly noncentral-F distribution. Another approximation in terms of a noncentral-F distribution is also proposed. Both these approximations are seen to compare favorably with some earlier approximations due to Das Gupta (1968) and Tiku (1972). The problem of approximating the cumulants of the doubly noncentral-F variable, which is pivotal in Tiku’s approximation, is examined and use of a noncentral-F distribution is seen to provide a good solution for it. A FORTRAN routine for the Edgeworth-series approximation is given.