Calculating the density and distribution function for the singly and doubly noncentral F

We derive a saddlepoint approximation to the pdf and cdf of both a singly and doubly noncentral F random variable and demonstrate its high accuracy over the entire parameter space. In contrast to usual saddlepoint applications, the method admits a closed form solution implying that, particularly for the doubly noncentral case, the proposed method is several orders of magnitude faster than existing computational methods. We also draw attention to existing errors in popular software packages, prove uniformity of error of the approximation, and provide an example demonstrating use of the new method for sample size calculation in experimental design. Keywords|Design of Experiments; Ratios of Quadratic Forms in Normal Variables; Saddlepoint Approximation; Statistical Computing. Research supported in part by NSF grant DMS-9625396. Corresponding author. e-mail: [email protected]