## The Annals of Statistics

### 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

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