Location estimator ($\hat{\theta }$)

A location estimator estimates the common shift in a location family, for example observations $X_i=\theta +{\epsilon }_i$ with a fixed error probability distribution. A translation-equivariant estimator obeysChanging the origin of measurement then shifts the estimate by the same amount. Both the sample mean and sample median have this property, but their sensitivity to an outlier differs.

For observations $0,1,2$, both estimates equal $1$. Replacing $2$ by $100$ changes the sample mean to $101/3$ while leaving the sample median at $1$. This illustrates resistance to an extreme observation, not a claim that the sample median is optimal under every probability distribution. A location parameter need not be a population mean: a Cauchy distribution has a location parameter although its expected value does not exist.