Computing the noncentral-F distribution and the power of the F-test with guaranteed accuracy

Abstract

The computations involving the noncentral-F distribution are notoriously difficult to implement properly in floating-point arithmetic: Catastrophic loss of precision, floating-point underflow and overflow, drastically increasing computation time and program hang-ups, and instability due to numerical cancellation have all been reported. It is therefore recommended that existing statistical packages are crosschecked, and the present paper proposes a numerical algorithm precisely for this purpose. To the best of our knowledge, the proposed method is the first method that can compute the noncentrality parameter of the noncentral-F distribution with guaranteed accuracy over a wide parameter range that spans the range relevant for practical applications. Although the proposed method is limited to cases where the the degree of freedom of the denominator of the F test statistic is even, it does not affect its usefulness significantly: All of those algorithmic failures and inaccuracies that we can still reproduce today could have been prevented by simply cross-checking against the proposed method. Two numerical examples are presented where the intermediate computations went wrong silently, but the final result of the computations seemed nevertheless plausible, and eventually erroneous results were published. Cross-checking against the proposed method would have caught the numerical errors in both cases. The source code of the algorithm is available on GitHub, together with self-contained command-line executables. These executables can read the data to be cross-checked from plain text files, making it easy to cross-check any statistical software in an automated fashion.

On the computation and inversion of the cumulative noncentral beta distribution function

“…For z (1) = 0.4, giving a positive ζ 0 , the zero that is larger than x 0 is an approximation of x, because when B p,q (x, y) is smaller than 0.5, x is larger than x 0 (see (6.13)). For z (2) = 0.6 the corresponding x is smaller than x 0 .…”

“…In [4], [1] and [2] the definition of the noncentral beta function is 6) and the relation with our definition is…”

“…The x-values resulting in the inversion of (7.3) for these z-values are x (1) . = 7.1704, x (2) . = 2.1475, respectively.…”

“…= 2.1475, respectively. In order to check the accuracy of the approximations, we can compute the values of the noncentral beta function for x (1) and x (2) : B 10,15 (x (1) , 0.45) . = 0.40900, B 10,15 (x (2) , 0.45) .…”

Bayesian Analysis on a Noncentral Fisher–Student’s Hypersphere

“…The latter distribution is known to generate numerical instabilities. Indeed, Baharev, Schichl, and Rév (2017) (33) for the noncentral hypersphere distribution should help avoiding such numerical instabilities, while allowing easier exploitation of the Bayesian hypothesis testing framework as was carried out in Section 4. Finally, note that definitions of the noncentrality parameter vary in the literature: numerical computations as carried in Figure 10 indicates that equation (25) refers to the parameter λ (herein ) used by Walck (2007).…”