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 $X$ 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.^[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 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.

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.ds.unifi.it/VL/VL_EN/special/special1.html

Probability distributions

List of probability distributions

Discrete univariate with finite support	Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta 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 Flory-Schulz Fréchet Gamma Gamma/Gompertz 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 matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson 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 extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet 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

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