Location–scale family

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 ${\displaystyle X}$ whose probability distribution function belongs to such a family, the distribution function of ${\displaystyle Y{\stackrel {d}{=}}a+bX}$ also belongs to the family (where ${\displaystyle {\stackrel {d}{=}}}$ means "equal in distribution"—that is, "has the same distribution as"). Moreover, if ${\displaystyle X}$ and ${\displaystyle Y}$ are two random variables whose distribution functions are members of the family, and assuming

1. existence of the first two moments and
2. ${\displaystyle X}$ has zero mean and unit variance,

then ${\displaystyle Y}$ can be written as ${\displaystyle Y{\stackrel {d}{=}}\mu _{Y}+\sigma _{Y}X}$ , where ${\displaystyle \mu _{Y}}$ and ${\displaystyle \sigma _{Y}}$ are the mean and standard deviation of ${\displaystyle Y}$.

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

• If ${\displaystyle X}$ has a cumulative distribution function ${\displaystyle F_{X}(x)=P(X\leq x)}$, then ${\displaystyle Y{=}a+bX}$ has a cumulative distribution function ${\displaystyle F_{Y}(y)=F_{X}\left({\frac {y-a}{b}}\right)}$.
• If ${\displaystyle X}$ is a discrete random variable with probability mass function ${\displaystyle p_{X}(x)=P(X=x)}$, then ${\displaystyle Y{=}a+bX}$ is a discrete random variable with probability mass function ${\displaystyle p_{Y}(y)=p_{X}\left({\frac {y-a}{b}}\right)}$.
• If ${\displaystyle X}$ is a continuous random variable with probability density function ${\displaystyle f_{X}(x)}$, then ${\displaystyle Y{=}a+bX}$ is a continuous random variable with probability density function ${\displaystyle f_{Y}(y)={\frac {1}{b}}f_{X}\left({\frac {y-a}{b}}\right)}$.

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.[1][2][3]

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 a generalized Student's t-distribution with 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 of 1.

References

1. Meyer, Jack (1987). "Two-Moment Decision Models and Expected Utility Maximization". American Economic Review. 77 (3): 421–430. JSTOR 1804104.
2. Mayshar, J. (1978). "A Note on Feldstein's Criticism of Mean-Variance Analysis". Review of Economic Studies. 45 (1): 197–199. JSTOR 2297094.
3. Sinn, H.-W. (1983). Economic Decisions under Uncertainty (Second English ed.). North-Holland.