Scale parameter

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