Location parameter: Difference between revisions

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	:<math>f_{\mu,\theta}(x) = f_\theta(x-\mu)</math>		:<math>f_{\mu,\theta}(x) = f_\theta(x-\mu)</math>
	where ''μ'' is the location parameter, ''θ'' represents additional parameters, and <math>f_\theta</math> is a function of the additional parameters.		where ''μ'' is the location parameter, ''θ'' represents additional parameters, and <math>f_\theta</math> is a function of the additional parameters.

			On the ] estimator of the location parameter see <ref> 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. </ref> .



	==Additive noise==		==Additive noise==
	An alternative way of thinking of location families is through the concept of ]. If ''μ'' is an unknown constant and ''w'' is random ] with probability density <math>f(w)</math>, then <math>x = \mu + w</math> has probability density <math>f_\mu(x) = f(x-\mu)</math> and is therefore a location family.		An alternative way of thinking of location families is through the concept of ]. If ''μ'' is an unknown constant and ''w'' is random ] with probability density <math>f(w)</math>, then <math>x = \mu + w</math> has probability density <math>f_\mu(x) = f(x-\mu)</math> and is therefore a location family.

			==References==
			{{Reflist}}

	==See also==		==See also==

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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

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

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

where μ is the location parameter, θ represents additional parameters, and $f_{\theta }$ 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 $f(w)$ , then $x=\mu +w$ has probability density $f_{\mu }(x)=f(x-\mu )$ and is therefore a location family.

References

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.

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Revision as of 18:38, 7 January 2011

Additive noise

References

See also