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On the computation of the noncentral F and noncentral beta distribution

机译：关于非中心F和非中心beta分布的计算

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摘要

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.
机译：不幸的是，计算非中心F和Beta分布的通信分布函数（cdf）和非中心参数的众多算法中的许多算法都可能产生完全错误的结果，如本文的示例所示。审查现有算法，并确定涉及数字困难的那些部分。结果，提出了伪代码，其中解决了所有已知的数值问题。无需理解复杂的数学背景，即可轻松地用编程语言C或FORTRAN实现此伪代码。提出了一个有限和封闭公式的符号评估，以计算精确的cdf值。这种方法可以在没有任何编程知识的情况下，在非常宽的参数范围内快速，可靠地检查由专业统计软件包返回的值。这项研究的动机是，一个非常有用的表用于计算方差分析表的可检测效应的大小，与通过Patnaik的2矩中心F逼近获得的值相比，包含较大的非中心性参数值区域中的可疑值。确定原因，并给出用于方差分析的表格更正形式。还讨论了非中心F分布的近似精度。

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