# Difference between revisions of "Scale parameter"

Jump to: navigation, search

A so-called positional parameter, which parametrizes a family of probability distributions of one type. A distribution in $\mathbf R$ with distribution function $F$ is said to belong to the same type as a fixed distribution with distribution function $F _ {0}$ if $F ( x) = F _ {0} (( x - b)/a)$. Here $a > 0$ is the scale parameter and $b$ is the shift parameter (or centralizing parameter). The meaning of the scale parameter is as follows: If $F _ {0}$ and $F$ are the distribution functions of random variables $X _ {0}$ and $X$, respectively, then a transition from $X _ {0}$ to $X = aX _ {0}$( for $b = 0$) represents a change in the unit of measurement.

#### Comments

The (possibly multi-dimensional) parameter $b$ in the family $F _ {0} ( ( x- b) / a )$ is also called the location parameter. The whole family of distributions is sometimes called a location-scale family of distributions.

#### References

 [a1] L. Breiman, "Statistics with a view towards applications" , Houghton Mifflin (1973) pp. 34–40
How to Cite This Entry:
Scale parameter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Scale_parameter&oldid=13206
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article