Statistics and Computing

, Volume 18, Issue 3, pp 333–340

# On the computation of the noncentral F and noncentral beta distribution

• Ali Baharev
• Sándor Kemény
Article

## Abstract

Unfortunately many of the numerous algorithms for computing the comulative distribution function (cdf) and noncentrality parameter of the noncentral F and beta distributions can produce 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 very useful table for calculating the size of detectable effects for ANOVA tables contains suspect values in the region of large noncentrality parameter values compared to the values obtained by Patnaik’s 2-moment central-F approximation. The cause 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.

## Keywords

Minimal detectable differences ANOVA Noncentrality parameter Central-F approximations to noncentral F Recursive algorithms Symbolic computation

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