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A so-called positional parameter, which parametrizes a family of probability distributions of one type. A distribution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s0832701.png" /> with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s0832702.png" /> is said to belong to the same type as a fixed distribution with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s0832703.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s0832704.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s0832705.png" /> is the scale parameter and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s0832706.png" /> is the shift parameter (or centralizing parameter). The meaning of the scale parameter is as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s0832707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s0832708.png" /> are the distribution functions of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s0832709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s08327010.png" />, respectively, then a transition from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s08327011.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s08327012.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s08327013.png" />) represents a change in the unit of measurement.
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A so-called positional parameter, which parametrizes a family of probability distributions of one type. A distribution in  $  \mathbf R $
 +
with distribution function  $  F $
 +
is said to belong to the same type as a fixed distribution with distribution function  $  F _ {0} $
 +
if  $  F ( x) = F _ {0} (( x - b)/a) $.
 +
Here  $  a > 0 $
 +
is the scale parameter and  $  b $
 +
is the shift parameter (or centralizing parameter). The meaning of the scale parameter is as follows: If  $  F _ {0} $
 +
and  $  F $
 +
are the distribution functions of random variables  $  X _ {0} $
 +
and  $  X $,
 +
respectively, then a transition from  $  X _ {0} $
 +
to  $  X = aX _ {0} $(
 +
for  $  b = 0 $)
 +
represents a change in the unit of measurement.
  
 
====Comments====
 
====Comments====
The (possibly multi-dimensional) parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s08327014.png" /> in the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083270/s08327015.png" /> is also called the location parameter. The whole family of distributions is sometimes called a location-scale family of distributions.
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The (possibly multi-dimensional) parameter $  b $
 +
in the family $  F _ {0} ( ( x- b) / a ) $
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is also called the location parameter. The whole family of distributions is sometimes called a location-scale family of distributions.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Breiman,  "Statistics with a view towards applications" , Houghton Mifflin  (1973)  pp. 34–40</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Breiman,  "Statistics with a view towards applications" , Houghton Mifflin  (1973)  pp. 34–40</TD></TR></table>

Latest revision as of 08:12, 6 June 2020


A so-called positional parameter, which parametrizes a family of probability distributions of one type. A distribution in $ \mathbf R $ with distribution function $ F $ is said to belong to the same type as a fixed distribution with distribution function $ F _ {0} $ if $ F ( x) = F _ {0} (( x - b)/a) $. Here $ a > 0 $ is the scale parameter and $ b $ is the shift parameter (or centralizing parameter). The meaning of the scale parameter is as follows: If $ F _ {0} $ and $ F $ are the distribution functions of random variables $ X _ {0} $ and $ X $, respectively, then a transition from $ X _ {0} $ to $ X = aX _ {0} $( for $ b = 0 $) represents a change in the unit of measurement.

Comments

The (possibly multi-dimensional) parameter $ b $ in the family $ F _ {0} ( ( x- b) / a ) $ is also called the location parameter. The whole family of distributions is sometimes called a location-scale family of distributions.

References

[a1] L. Breiman, "Statistics with a view towards applications" , Houghton Mifflin (1973) pp. 34–40
How to Cite This Entry:
Scale parameter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Scale_parameter&oldid=13206
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article