Location parameter

An alternative way of thinking of location families is through the concept of additive noise. If ${\displaystyle x_{0}}$ is a constant and W is random noise with probability density ${\displaystyle f_{W}(w),}$ then ${\displaystyle X=x_{0}+W}$ has probability density ${\displaystyle f_{x_{0}}(x)=f_{W}(x-x_{0})}$ and its distribution is therefore part of a location family.

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

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

it can be proved that ${\displaystyle g}$ is a p.d.f. by verifying if it respects the two conditions[4] ${\displaystyle g(x|\theta ,x_{0})\geq 0}$ and ${\displaystyle \int _{-\infty }^{\infty }g(x|\theta ,x_{0})dx=1}$. ${\displaystyle g}$ integrates to 1 because:

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

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

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

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

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