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.

In other words, when the function is graphed, 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 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

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