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Location parameter

In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter <math>x_0</math>, 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

<math>f_{x_0}(x) = f(x - x_0).</math>

Here, <math>x_0</math> is called the location parameter. Examples of location parameters include the mean, the median, and the mode.

Thus in the one-dimensional case 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 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.

Additive noise

An alternative way of thinking of location families is through the concept of additive noise. If <math>x_0</math> is a constant and W is random noise with probability density <math>f_W(w),</math> then <math>X = x_0 + W</math> has probability density <math>f_{x_0}(x) = f_W(x-x_0)</math> and its distribution is therefore part of a location family.

See also

de:Parameter (Statistik)#Lageparameter