= Translation-invariant estimator
= Translation-equivariant estimator
{synonym}
For a location <estimator>, the property historically called translation invariance in this setting is more explicitly called translation <equivariance>:
$$
T_n(x_1+a,\ldots,x_n+a)=T_n(x_1,\ldots,x_n)+a\qquad(a\in\mathbb R).
$$
Changing the origin of the measurements changes the estimated location by exactly the same amount. Both the <sample mean> and <sample median> satisfy this identity. For example, adding $10$ to every observation adds $10$ to either estimated location.
Strict invariance would instead leave the numerical answer unchanged. The <sample variance> has that different property because shifting every observation also shifts their <sample mean>, leaving all centered differences unchanged. The distinction matters when specifying the admissible <estimators> in a <minimax asymptotic bias> problem. The older translation-invariant terminology appears in <Peter J. Huber>'s original location analysis.<Robust Estimation of a Location Parameter by Peter J. Huber>[Introduction: translation-invariant location estimators]{ref}
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