Difference between revisions of "Scale parameter"
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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 | + | 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==== | ====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
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