Location-scale family

In probability theory, especially in mathematical statistics, a location-scale family is a family of univariate probability distributions parametrized by a location parameter and a non-negative scale parameter. For any random variable $X$ whose probability distribution function belongs to such a family, the distribution function of $Y \stackrel{d}{=} a + b X$ also belongs to the family (where $\stackrel{d}{=}$ means "equal in distribution"—that is, "has the same distribution as"). Moreover, if $X$ and $Y$ are two random variables whose distribution functions are members of the family, and $X$ has zero mean and unit variance, then $Y$ can be written as $Y \stackrel{d}{=} \mu_Y + \sigma_Y X$ , where $\mu_Y$ and $\sigma_Y$ are the mean and standard deviation of $Y$.

In other words, a class $\Omega$ of probability distributions is a location-scale family if for all cumulative distribution functions $F \in \Omega$ and any real numbers $a \in \mathbb{R}$ and $b > 0$, the distribution function $G(x) = F(a + b x)$ is also a member of $\Omega$.

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. Well-known families in which the functional form of the distribution is consistent throughout the family include the following:

• Normal distribution
• Elliptical distribution
• Cauchy distribution
• Uniform distribution (continuous)
• Uniform distribution (discrete)
• Logistic distribution
• Laplace distribution
• Student's t-distribution
• Generalized extreme value distribution

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.

Probability density function (PDF):	dt_ls(x, df, mu, sigma) =	1/sigma * dt((x - mu)/sigma, df)
Cumulative distribution function (CDF):	pt_ls(x, df, mu, sigma) =	pt((x - mu)/sigma, df)
Quantile function (inverse CDF):	qt_ls(prob, df, mu, sigma) =	qt(prob, df)*sigma + mu
Generate a random variate:	rt_ls(df, mu, sigma) =	rt(df)*sigma + mu

Note that the generalized functions do not have standard deviation sigma since the standard t distribution does not have standard deviation 1.

External links

• http://www.ds.unifi.it/VL/VL_EN/special/special1.html