Location parameter

In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter $x_{0}$, 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

$f_{x_{0}}(x)=f(x-x_{0}).$

Here, $x_{0}$ is called the location parameter. Examples of location parameters include the mean, the median, and the mode.

Thus in the one-dimensional case if $x_{0}$ is increased, the probability density or mass function shifts rigidly to the left, 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

$f_{x_{0},\theta }(x)=f_{\theta }(x-x_{0})$

where $x_{0}$ is the location parameter, θ represents additional parameters, and $f_{\theta }$ 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 $x_{0}$ is a constant and W is random noise with probability density $f_{W}(w),$ then $X=x_{0}+W$ has probability density $f_{x_{0}}(x)=f_{W}(x-x_{0})$ 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