Location parameter - Alchetron, The Free Social Encyclopedia

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 , θ ( x ) = f θ ( x − x 0 )

where x 0 is the location parameter, θ represents additional parameters, and f θ 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.

References

Location parameter Wikipedia

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