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On a new transformation to normality. (English) Zbl 0564.62012
A Gaussian approximation to the distribution of the non-negative random variable Y is developed using the E. B. Wilson and M. M. Hilferty [Proc. Natl. Acad. Sci. USA 17, 684-688 (1931; Zbl 0004.36005)] approach. This approximation uses the symmetrizing transformation $$((Y+b)/k_ 1)^ h$$ where $$k_ 1$$ is the first moment of Y and h and b are determined from the first three cumulants of Y.
The approximation is illustrated in the case in which Y is a non-central chi-square, where numerical evaluations indicate that the new transformation is an improvement over existing ones especially for small values of $$k_ 1$$.

##### MSC:
 6.2e+21 Asymptotic distribution theory in statistics 6.2e+100 Statistical distribution theory
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