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 $Y{\stackrel {d}{=}}a+bX$ 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 $G(x)=F(a+bx)$ 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

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[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.

Probability distributions (List)
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Fréchet gamma gamma/Gompertz generalized gamma generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared noncentral F Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped