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Location-Scale Distributions

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International Encyclopedia of Statistical Science

A random variable X with realization x belongs to the location-scale family when its cumulative distribution is a function only of \((x - a)/b\):

$${F}_{X}(x\,\vert \,a,b) = \text{ Pr}(X \leq x\vert a,b) = F\left ({x - a} {b} \right );\ a \in \mathbb{R},\ b > 0;$$

where F( ⋅) is a distribution having no other parameters. Different F( ⋅)’s correspond to different members of the family. (a, b) is called the location–scale parameter, a being the location parameter and b being the scale parameter. For fixed b = 1 we have a subfamily which is a location family with parameter a, and for fixed a = 0 we have a scale family with parameter b. The variable

$$Y = \frac{X - a}{b}$$

is called the reduced or standardized variable. It has a = 0 and b = 1. If the distribution of X is absolutely continuous with density function

$${f}_{X}(x\,\vert \,a,b) = \frac{\text{ d}{F}_{X}(x\,\vert \,a,b)}{\text{ d}\,x}$$

then (a, b) is a location scale-parameter for the distribution of X if (and only if)

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References and Further Reading

  • Barnett V (1975) Probability plotting methods and order statistics. Appl Stat 24:95–108

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  • Barnett V (1976) Convenient probability plotting positions for the normal distribution. Appl Stat 25:47–50

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  • Blom G (1958) Statistical estimates and transformed beta variables. Almqvist and Wiksell, Stockholm

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  • Kimball BF (1960) On the choice of plotting positions on probability paper. J Am Stat Assoc 55:546–560

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  • Lloyd EH (1952) Least-squares estimation of location and scale parameters using order statistics. Biometrika 39:88–95

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  • Mi J (2006) MLE of parameters of location–scale distributions for complete and partially grouped data. J Stat Planning Inference 136:3565–3582

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  • Rinne H (2010) Location-scale distributions – linear estimation and probability plotting; http:geb.uni-giessen/geb/volltexte/2010/7607/

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Author information

Authors and Affiliations

  1. Justus–Liebig–University Giessen, Giessen, Germany

    Horst Rinne (Professor Emeritus for Statistics and Econometrics)

Authors
  1. Horst Rinne
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Editor information

Editors and Affiliations

  1. Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia

    Miodrag Lovric

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© 2011 Springer-Verlag Berlin Heidelberg

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Rinne, H. (2011). Location-Scale Distributions. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_341

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