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 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

$f_{x_0,\theta}(x) = f_\theta(x-x_0)$

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