## The Annals of Statistics

- Ann. Statist.
- Volume 10, Number 2 (1982), 323-339.

### Transformation Theory: How Normal is a Family of Distributions?

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 10, Number 2 (1982), 323-339.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345777

**Digital Object Identifier**

doi:10.1214/aos/1176345777

**Mathematical Reviews number (MathSciNet)**

MR653511

**Zentralblatt MATH identifier**

0507.62008

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E10: Characterization and structure theory

Secondary: 62E99: None of the above, but in this section

**Keywords**

Normalization variance stabilization square root transformation power transformations

#### Citation

Efron, Bradley. Transformation Theory: How Normal is a Family of Distributions?. Ann. Statist. 10 (1982), no. 2, 323--339. doi:10.1214/aos/1176345777. https://projecteuclid.org/euclid.aos/1176345777

#### Corrections

- See Correction: Bradley Efron. Correction: "Transformation Theory: How Normal is a Family of Distributions. Ann. Statist., vol. 10, no. 3 (1982), 1032.Project Euclid: euclid.aos/1176345897