On the computation of the noncentral F and noncentral beta distribution

Abstract

); and Sándor Kemény is professor at the same institution (email: kemeny@mail.bme.hu ). The authors wish to thank Mr. Richárd Király for his preliminary work. The authors are grateful to the Associate Editor of STCO and the unknown reviewers for their helpful suggestions. 2 ABSTRACT AND KEY WORDSUnfortunately many of the numerous algorithms for computing the cdf and noncentrality parameter of the noncentral F and beta distributions can return completely incorrect results as demonstrated in the paper by examples. Existing algorithms are scrutinized and those parts that involve numerical difficulties are identified. As a result, a pseudo code is presented in which all the known numerical problems are resolved. This pseudo code can be easily implemented in programming language C or FORTRAN without understanding the complicated mathematical background. Symbolic evaluation of a finite and closed formula is proposed to compute exact cdf values. This approach makes it possible to check quickly and reliably the values returned by professional statistical packages over an extraordinarily wide parameter range without any programming knowledge. This research was motivated by the fact that a most useful table for calculating the size of detectable effects for ANOVA tables contains suspicious values in the region of large noncentrality parameter values compared to the values obtained by Patnaik's 2-moment central-F approximation. The reason is identified and the corrected form of the table for ANOVA purposes is given. The accuracy of the approximations to the noncentral-F distribution is also discussed.

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

“…-Under-and overflow problems were reported by Benton and Krishnamoorthy (2003), Ding (1997), Helstrom and Ritcey (1985) independently of us; -we also reveal in Baharev and Kemény 2008 that the algorithms of Norton (1983) and Lenth (1987) are exposed to over-and underflow issues, and that the Appendix 12 of Lorenzen and Anderson (1993, p. 374) is most likely bogus due to overflow; -catastrophic round-off errors were reported by Frick (1990); -drastically increasing computation time and hang-ups were observed by Chattamvelli (1995), Benton and Krishnamoorthy (2003); -other noncentral distributions are similarly challenging, see for example Oliveira and Ferreira (2012). If some of the intermediate computations suffer catastrophic loss of precision, the root-finding method can still succeed, and the final result presented to the user may nevertheless seem plausible.…”

“…In both examples, the intermediate computations suffer significant loss of precision, but the final results presented to the user seem nevertheless plausible, making these kinds of numerical errors particularly harmful. A by-product of the first example is that the algorithm of Baharev and Kemény (2008), implemented on the top of the built-in functions of R, is proved to be accurate for 6 significant digits in the investigated cases with mathematical certainty.…”

“…All entries in Appendix 12 seem plausible; there is no obvious sign that some of the entries have no correct significant digits (for example that entry in Appendix 12 that corresponds to the top right entry of Table 4 of the present paper). The entries of Appendix 12 are most likely bogus due to floating-point overflow (Baharev and Kemény 2008).…”

“…The corrected form of Appendix 12 of Lorenzen and Anderson (1993, p. 374), i.e., Table 3 of Baharev and Kemény (2008), fully spans the parameter range that is relevant for practical applications. This table (except rows for which b is not integer, i.e., b = 0.5, 1.5, 2.5, 3.5) is recomputed with the proposed method, and it is given as Table 4.…”

“…These methods were published more than 20 years ago, and their source code is not publicly available. As we discuss in Baharev and Kemény (2008), the method in Wang and Kennedy (1995) is susceptible to over-and underflow: For example, it would over-or underflow due to the large value of the noncentrality parameter if we tried to compute the top right entry of Table 4 in contrast to the proposed method that has no difficulties there.…”