Location parameter

In statistics, a location family is a class of probability distributions that is 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. Examples of location parameters include the mean, the median, and the mode.

Thus if $\mu$ 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 families. In this case, the probability density function or probability mass function will be a special case of the more general 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.

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 $f_{W}(w)$, then $X=\mu +W$ has probability density $f_{\mu }(x)=f_{W}(x-\mu )$ and is therefore a location family.

See also

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