Location parameter

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

Here, is called the location parameter. Examples of location parameters include the mean, the median, and the mode.

Thus in the one-dimensional case if 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

where is the location parameter, θ represents additional parameters, and is a function parametrized on the additional parameters.

Additive noise[edit]

An alternative way of thinking of location families is through the concept of additive noise. If is a constant and W is random noise with probability density then has probability density and its distribution is therefore part of a location family.

See also[edit]