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 {displaystyle 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 In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter x 0 {displaystyle 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 Here, x 0 {displaystyle 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 {displaystyle 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 where x 0 {displaystyle x_{0}} is the location parameter, θ represents additional parameters, and f θ {displaystyle f_{ heta }} is a function parametrized on the additional parameters. An alternative way of thinking of location families is through the concept of additive noise. If x 0 {displaystyle x_{0}} is a constant and W is random noise with probability density f W ( w ) , {displaystyle f_{W}(w),} then X = x 0 + W {displaystyle X=x_{0}+W} has probability density f x 0 ( x ) = f W ( x − x 0 ) {displaystyle f_{x_{0}}(x)=f_{W}(x-x_{0})} and its distribution is therefore part of a location family.