Misplaced Pages

This is an old revision of this page, as edited by Gjshisha (talk | contribs) at 18:38, 7 January 2011 (a new reference was inserted). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Location parameter" – news · newspapers · books · scholar · JSTOR (December 2009) (Learn how and when to remove this message)

In statistics, a location family is a class of probability distributions parametrized by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution. Formally, this means that the probability density functions or probability mass functions in this class have the form

${\displaystyle f_{\mu }(x)=f(x-\mu ).}$

Here, μ is called the location parameter.

In other words, when you graph the function, the location parameter determines where the origin will be located. If μ is positive, the origin will be shifted to the right, and if μ is negative, it will be shifted to the left.

A location parameter can also be found in families having more than one parameter, such as location-scale families. In this case, the probability density function or probability mass function will have the form

${\displaystyle f_{\mu ,\theta }(x)=f_{\theta }(x-\mu )}$

where μ is the location parameter, θ represents additional parameters, and ${\displaystyle f_{\theta }}$ is a function of the additional parameters.

On the maximum likelihood estimator of the location parameter see .

Additive noise

An alternative way of thinking of location families is through the concept of additive noise. If μ is an unknown constant and w is random noise with probability density ${\displaystyle f(w)}$, then ${\displaystyle x=\mu +w}$ has probability density ${\displaystyle f_{\mu }(x)=f(x-\mu )}$ and is therefore a location family.

References

1. Székely, G. J. and Buczolich, Z. (1989) When is a weighted average of ordered sample elements a maximum likelihood estimator of the location parameter? Advances in Applied Mathematics 10, 439-456.

See also

• Location test
• Invariant estimator
• Location-scale family
• Scale parameter

• Summary statistics
• Theory of probability distributions
• Statistical terminology