en>Bryard at 10:46, 9 January 2015

2015-01-09T10:46:22Z

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en>Justgravy at 00:13, 22 February 2014

2014-02-22T00:13:56Z

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en>SashaBarks: /* Early life */ This is less ambiguous

2014-01-25T14:32:14Z

Early life: This is less ambiguous

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{{Unreferenced|date=December 2009}}
In [[statistics]], a '''location family''' is a class of [[probability distribution]]s that is [[statistical parameter|parametrized]] by a scalar- or vector-valued parameter <math>x_0</math>, which determines the "location" or shift of the distribution. Formally, this means that the [[probability density function]]s or [[probability mass function]]s in this class have the form
:<math>f_{x_0}(x) = f(x - x_0).</math>
Here, <math>x_0</math> is called the '''location parameter'''. Examples of location parameters include the [[mean (statistics)|mean]], the [[median]], and the [[Mode (statistics)|mode]].

Thus in the one-dimensional case if <math>x_0</math> is increased, the probability density or mass function shifts rigidly to the right, maintaining its exact shape.

A location parameter can also be found in families having more than one parameter, such as [[location-scale family|location-scale families]]. In this case, the probability density function or probability mass function will be a special case of the more general form
:<math>f_{x_0,\theta}(x) = f_\theta(x-x_0)</math>
where <math>x_0</math> is the location parameter, ''θ'' represents additional parameters, and <math>f_\theta</math> is a function parametrized on the additional parameters.

==Additive noise==
An alternative way of thinking of location families is through the concept of [[additive noise]]. If <math>x_0</math> is an constant and ''W'' is random [[noise]] with probability density <math>f_W(w),</math> then <math>X = x_0 + W</math> has probability density <math>f_{x_0}(x) = f_W(x-x_0)</math> and its distribution is therefore part of a location family.

==See also==
* [[Central tendency]]
* [[Location test]]
* [[Invariant estimator]]
* [[Location-scale family]]
* [[Scale parameter]]
* [[Two-moment decision models]]

{{Statistics|descriptive|state=collapsed}}

{{DEFAULTSORT:Location Parameter}}
[[Category:Summary statistics]]
[[Category:Theory of probability distributions]]
[[Category:Statistical terminology]]

[[de:Parameter (Statistik)#Lageparameter]]

Kate Bush - Revision history

en>Bryard at 10:46, 9 January 2015

en>Justgravy at 00:13, 22 February 2014

en>SashaBarks: /* Early life */ This is less ambiguous