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 $X$ 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 assuming

existence of the first two moments 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 $\Omega$ .

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

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

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

External links

http://www.randomservices.org/random/special/LocationScale.html

[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.

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