Editing Location parameter (section)

{{Short description|Concept in statistics}}
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In [[statistics]], a '''location parameter''' of a [[probability distribution]] is a scalar- or vector-valued [[statistical parameter|parameter]] <math>x_0</math>, which determines the "location" or shift of the distribution. In the literature of location parameter estimation, the probability distributions with such parameter are found to be formally defined in one of the following equivalent ways:
* either as having a [[probability density function]] or [[probability mass function]] <math>f(x - x_0)</math>;<ref>{{cite journal |last1=Takeuchi |first1=Kei |title= A Uniformly Asymptotically Efficient Estimator of a Location Parameter |journal=Journal of the American Statistical Association |date=1971 |volume=66 |issue=334 |pages=292–301|doi=10.1080/01621459.1971.10482258 |s2cid=120949417 }}</ref> or
* having a [[cumulative distribution function]] <math>F(x - x_0)</math>;<ref>{{cite book |last1=Huber |first1=Peter J. |chapter=Robust Estimation of a Location Parameter |title=Breakthroughs in Statistics |series=Springer Series in Statistics |date=1992 |pages=492–518| publisher=Springer|doi=10.1007/978-1-4612-4380-9_35 |isbn=978-0-387-94039-7 |chapter-url=http://projecteuclid.org/euclid.aoms/1177703732 }}</ref> or
* being defined as resulting from the random variable transformation <math>x_0 + X</math>, where <math>X</math> is a random variable with a certain, possibly unknown, distribution.<ref>{{cite journal |last1=Stone |first1=Charles J. |title=Adaptive Maximum Likelihood Estimators of a Location Parameter |journal=The Annals of Statistics |date=1975 |volume=3 |issue=2 |pages=267–284|doi=10.1214/aos/1176343056 |doi-access=free }}</ref> See also {{Slink||Additive noise}}.

A direct example of a location parameter is the parameter <math>\mu</math> of the [[normal distribution]]. To see this, note that the probability density function <math>f(x | \mu, \sigma)</math> of a normal distribution <math>\mathcal{N}(\mu,\sigma^2)</math> can have the parameter <math>\mu</math> factored out and be written as: 
:<math>
g(x' = x - \mu | \sigma) = \frac{1}{\sigma \sqrt{2\pi} } \exp(-\frac{1}{2}\left(\frac{x'}{\sigma}\right)^2)
</math>
thus fulfilling the first of the definitions given above.

The above definition indicates, in the one-dimensional case, that 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.