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.

A direct example of location parameter is the parameter $\mu$ of the normal distribution. To see this, note that the p.d.f. (probability density function) of a normal ${\mathcal {N}}(0,\sigma ^{2})$ is given by

$f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x}{\sigma }}\right)^{2}}$

and defining $g(x)=f(x-\mu )$, so $\mu$ is a location parameter according to the above definition, yields

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

which is precisely the p.d.f. of a normal ${\mathcal {N}}(\mu ,\sigma ^{2})$.

The above definition indicates, in the one-dimensional case, that 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.

Proofs

For the continuous univariate case, consider a probability density function $f(x|\theta ),x\in [a,b]\subset \mathbb {R}$, where $\theta$ is a vector of parameters. A location parameter $x_{0}$ can be added by defining:

$g(x|\theta ,x_{0})=f(x-x_{0}|\theta ),\;x\in [a-x_{0},b-x_{0}]$

it can be proved that $g$ is a p.d.f. by verifying if it respects the two conditions $g(x|\theta ,x_{0})\geq 0$ and $\int _{-\infty }^{\infty }g(x|\theta ,x_{0})dx=1$ . $g$ integrates to 1 because:

$\int _{-\infty }^{\infty }g(x|\theta ,x_{0})dx=\int _{a-x_{0}}^{b-x_{0}}g(x|\theta ,x_{0})dx=\int _{a-x_{0}}^{b-x_{0}}f(x-x_{0}|\theta )dx$

now making the variable change $u=x-x_{0}$ and updating the integration interval accordingly yields:

$\int _{a}^{b}f(u|\theta )du=1$

because $f(x|\theta )$ is a p.d.f. by hypothesis. $g(x|\theta ,x_{0})\geq 0$ follows from $g$ sharing the same image of $f$, which is a p.d.f. so its image is contained in $[0,1]$.

See also

• Central tendency
• Location test
• Invariant estimator
• Scale parameter
• Two-moment decision models