Location-scale family
In probability theory, especially as that field is used in statistics, a location-scale family is a family of univariate probability distributions parametrized by a location parameter and a non-negative scale parameter; if X is any random variable whose probability distribution belongs to such a family, then Y =d (a + bX) is another (where =d means "is equal in distribution to" — that is, "has the same distribution as"), and every distribution in the family is of that form. Moreover, if X is a zero-mean, unit-variance member of the family, then every member Y of the family can be written as Y =d (μY + σYX), where μY and σY are the mean and standard deviation of Y.
In other words, a class Ω of probability distributions is a location-scale family if whenever F is the cumulative distribution function of a member of Ω and a is any real number and b > 0, then G(x) = F(a + bx) is also the cumulative distribution function of a member of Ω.
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]
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Examples [edit]
In general, any random variable X is a member of a location-scale family with an infinite number of members; each of these members Y is distributed the same as a + bX for real numbers a and b > 0. However, the form of the distribution function for a + bX, and hence for Y, may differ from the form of the distribution function for X. Well-known families in which the functional form of the distribution is retained throughout the family include the following:
- Normal distribution
- Elliptical distribution
- 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 [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]
- ^ Meyer, Jack. "Two-moment decision models and expected utility maximization," American Economic Review 77, June 1987, 421–430.
- ^ Mayshar, J., "A note on Feldstein's criticism of mean-variance analysis," Review of Economic Studies 45, 1978, 197–199.
- ^ 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
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