In probability theory, especially in mathematical statistics, a location-scale family is a family of probability distributions parametrized by a location parameter and a non-negative scale parameter. For any random variable
Contents
In other words, a class
In decision theory, if all alternative distributions available to a decision-maker are in the same location-scale family, and the first two moments are finite, then a two-moment decision model can apply, and decision-making can be framed in terms of the means and the variances of the distributions.
Examples
Often, location-scale families are restricted to those where all members have the same functional form. Most location-scale families are univariate, though not all. Well-known families in which the functional form of the distribution is consistent throughout the family include the following:
Converting a single distribution to a location-scale family
The following shows how to implement a location-scale family in a statistical package or programming environment where only functions for the "standard" version of a distribution are available. It is designed for R but should generalize to any language and library.
The example here is of the Student's t-distribution, which is normally provided in R only in its standard form, with a single degrees of freedom parameter df
. The versions below with _ls
appended show how to generalize this to encompass an arbitrary location parameter mu
and scale parameter sigma
.
Note that the generalized functions do not have standard deviation sigma
since the standard t distribution does not have standard deviation.