Open Access
June, 1982 Transformation Theory: How Normal is a Family of Distributions?
Bradley Efron
Ann. Statist. 10(2): 323-339 (June, 1982). DOI: 10.1214/aos/1176345777


This paper concerns the following question: if $X$ is a real-valued random variate having a one-parameter family of distributions $\mathscr{F}$, to what extent can $\mathscr{F}$ be normalized by a monotone transformation? In other words, does there exist a single transformation $Y = g(X)$ such that $Y$ has, nearly, a normal distribution for every distribution of $X$ in $\mathscr{F}$? The theory developed answers the question without considering the form of $g$ at all. In those cases where the answer is positive, simple formulas for calculating $g$ are given. The paper also considers the relationship between normalization and variance stabilization.


Download Citation

Bradley Efron. "Transformation Theory: How Normal is a Family of Distributions?." Ann. Statist. 10 (2) 323 - 339, June, 1982.


Published: June, 1982
First available in Project Euclid: 12 April 2007

zbMATH: 0507.62008
MathSciNet: MR653511
Digital Object Identifier: 10.1214/aos/1176345777

Primary: 62E10
Secondary: 62E99

Keywords: normalization , power transformations , square root transformation , variance stabilization

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 2 • June, 1982
Back to Top