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Location parameter

 

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

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