Location–scale family explained

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").

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 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\le x) , then Y{ = } a + b X has a cumulative distribution function F _{Y(y)} = F _{X\left( (y - a)/(b)} \right) .

• If

X is a discrete random variable with probability mass function p _{X(x)=} P(X = x) , then Y{ = } a + b X is a discrete random variable with probability mass function p _{Y(y)} = p _{X\left( (y - a)/(b)} \right) .

• If

X is a continuous random variable with probability density function f _{X(x)} , then Y{ = } a + b X is a continuous random variable with probability density function f _{Y(y)} = (1)/(b) f _{X\left( (y - a)/(b)} \right) .

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 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:

• Normal distribution
• Elliptical distributions
• 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 a generalized Student's t-distribution with an arbitrary location parameter m and scale parameter s.

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

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

External links

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

Notes and References

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