zbMATH — the first resource for mathematics

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 \(k_ 1\).

62E20 Asymptotic distribution theory in statistics
62E99 Statistical distribution theory
Full Text: DOI
[1] Abdel-Aty S.H., Biometrika 41 pp 538– (1954)
[2] Fisher R.A., JRSS, Ser A 85 pp 87– (1922)
[3] Jensen D.R., JASA 67 pp 898– (1972)
[4] Johnson N.L., Continuoiis Univariate Distributions-1
[5] Moschopoulos P.G., Comm. Stat. 12 (1983)
[6] Mudholkar G.S., JASA 76 (374) pp 479– (1981) · doi:10.1080/01621459.1981.10477673
[7] Patnaik P.B., Biometrika 36 (374) pp 202– (1949)
[8] Pearson E.S., Biometrika 46 (374) pp 364– (1959) · Zbl 0101.35806 · doi:10.2307/2333533
[9] Wilson E.P., Proceedings of the National Academy of Science 17 (374) pp 684– (1931) · Zbl 0004.36005 · doi:10.1073/pnas.17.12.684
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.