Location scales family

In statistics  , a position-scale family or position and scale family is a family of probability distributions parameterized by a position parameter and a non-negative scale parameter .

definition

Let be a real random variable with distribution function , and for and be ${\displaystyle Z}$ ${\displaystyle F_{Z}}$ ${\displaystyle \mu \in \mathbb {R} }$ ${\displaystyle \sigma >0}$

${\displaystyle X=\mu +\sigma Z}$.

The resulting family of distributions is called an induced position-scale family ${\displaystyle Z}$ with position parameters and scale parameters . For one speaks of a pure scale family . ${\displaystyle \mu }$ ${\displaystyle \sigma }$ ${\displaystyle \mu =0}$

If on continuous and strictly monotonic, then the distribution function of on is also continuous and strictly monotonic and the following applies: ${\displaystyle F_{Z}}$ ${\displaystyle \{x\in \mathbb {R} :F_{Z}(x)\in (0,1)\}}$ ${\displaystyle F_{X}}$ ${\displaystyle X}$ ${\displaystyle \{x\in \mathbb {R} :F_{X}(x)\in (0,1)\}}$

${\displaystyle F_{X}^{-1}(u)=\mu +\sigma F_{Z}^{-1}(u),\;u\in (0,1)}$.

In the case of a pure scale family, the following applies

${\displaystyle F_{X}^{-1}(u)=\sigma F_{Z}^{-1}(u),\;u\in (0,1)}$.

Individual evidence

1. ↑ Torsten Becker, et al .: Stochastic risk modeling and statistical methods. Springer Spectrum, 2016. p. 357.
2. ↑ location-scale family. Glossary of statistical terms. In: International Statistical Institute . June 1, 2011, accessed May 19, 2020 . 
3. ↑ Torsten Becker, et al .: Stochastic risk modeling and statistical methods. Springer Spectrum, 2016. p. 357.
4. ↑ Torsten Becker, et al .: Stochastic risk modeling and statistical methods. Springer Spectrum, 2016. p. 357.