Location parameter

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In statistics, a location parameter of a probability distribution is a scalar- or vector-valued parameter ${\displaystyle x_{0}}$, which determines the "location" or shift of the distribution. In the literature of location parameter estimation, the probability distributions with such parameter are found to be formally defined in one of the following equivalent ways:

• either as having a probability density function or probability mass function ${\displaystyle f(x-x_{0})}$;^[1] or
• having a cumulative distribution function ${\displaystyle F(x-x_{0})}$;^[2] or
• being defined as resulting from the random variable transformation ${\displaystyle x_{0}+X}$, where ${\displaystyle X}$ is a random variable with a certain, possibly unknown, distribution^[3] (See also #Additive_noise).

A direct example of a location parameter is the parameter ${\displaystyle \mu }$ of the normal distribution. To see this, note that the probability density function ${\displaystyle f(x|\mu ,\sigma )}$ of a normal distribution ${\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}$ can have the parameter ${\displaystyle \mu }$ factored out and be written as:

${\displaystyle g(y-\mu |\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {y}{\sigma }}\right)^{2}}}$

thus fulfilling the first of the definitions given above.

The above definition indicates, in the one-dimensional case, that if ${\displaystyle x_{0}}$ 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

${\displaystyle f_{x_{0},\theta }(x)=f_{\theta }(x-x_{0})}$

where ${\displaystyle x_{0}}$ is the location parameter, θ represents additional parameters, and ${\displaystyle f_{\theta }}$ is a function parametrized on the additional parameters.