# 胜过 Fisher z 变换！（1）

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我们发现了两个新的初等显式函数，在逼近标准正态分布累积分布函数时，误差小于著名的 Fisher z 变换。
Fisher z 变换（Fisher z-transform，Fisher z transformation，Fisher's z' transformation，Fisher's Z Transformation）在当前国内外数理统计学教材中普遍使用。Fisher在1915年提出该变换。
Transactions of Tianjin University）录用。预期在2013年下半年刊出。
另外一个新函数还在争取发表的过程中。

TTU录用新函数逼近标准正态分布累积分布函数的误差如下图。其中红色细线为 Fisher z 变换的逼近误差，蓝色细线为我们的新函数。两者优化配合的误差为绿色点线。由于版权等科技规范的限制，这里我们不能提供有关的量化细节，只能提供直观的定性对照。

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————————— 相关背景 —————————

Sir Ronald Aylmer Fisher

Photograph courtesy of Professor A W F Edwards by kind permission of Joan Fisher Box

在《英百科全书Encyclopaedia Britannica
http://www.britannica.com/EBchecked/topic/208658/Sir-Ronald-Aylmer-Fisher

Sir Ronald Aylmer Fisher, byname R.A. Fisher    (born February 17, 1890, London, England—died July 29, 1962, Adelaide, Australia), British statistician and geneticist who pioneered the application of statistical procedures to the design of scientific experiments.

http://www-history.mcs.st-andrews.ac.uk/Biographies/Fisher.html

Fisher z-transform 在《苏联数学百科全书》的当前网络版

http://www.encyclopediaofmath.org/index.php/Correlation_(in_statistics)

If   one usually uses the Fisher z-transform, with   replaced by z according to the formula

Even at relatively small values the distribution of is a good approximation to the normal distribution with mathematical expectation

and variance . On this basis one can now define approximate confidence intervals for the true correlation coefficient .

For the distribution of the sample correlation ratio and for tests of the linearity hypothesis for the regression, see [3].

References

 [1] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) [2] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957) [3] M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2. Inference and relationship , Griffin (1979) [4] S.A. Aivazyan, "Statistical research on dependence" , Moscow (1968) (In Russian)

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[2] 毛西德格的瓷器密码：神马乖乖？
http://blog.sciencenet.cn/blog-107667-518968.html

Quadratic radical function better than fisher z transformation.Transactions of Tianjin University, October 2013, Volume 19, Issue 5, pp 381-384.

http://blog.sciencenet.cn/blog-107667-603297.html

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