Location-scale family

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In probability theory, especially in mathematical statistics, a location-scale family is a family of univariate 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, then a two-moment decision model applies, and decision-making can be framed in terms of the means and the variances of the distributions.[1][2][3]

Examples[edit]

Often, location-scale families are restricted to those where all members have the same functional form. 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[edit]

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[edit]

  1. ^ Meyer, Jack. "Two-moment decision models and expected utility maximization," American Economic Review 77, June 1987, 421–430.
  2. ^ Mayshar, J., "A note on Feldstein's criticism of mean-variance analysis," Review of Economic Studies 45, 1978, 197–199.
  3. ^ Sinn, H.-W., Economic Decisions under Uncertainty, second English edition, 1983, North-Holland.

Further references[edit]

http://www.ds.unifi.it/VL/VL_EN/special/special1.html