Location- and Scale-Invariant Statistic

Definition

Let $X_1,X_2,\dots ,X_n$ be a Random Sample from a location-scale model, and $Z=u(X_1,X_2,\dots ,X_n)$ be a statistic such that $\forall d\in \mathbb{R},\forall c\in {\mathbb{R}}^{+},u(c{x}_1+d,c{x}_2+d,\dots ,c{x}_n+d)=u(x_1,x_2,\dots ,x_n)$, then $Z=u({\theta }_1+{\theta }_2{W}_1,{\theta }_1+{\theta }_2{W}_2,\dots ,{\theta }_1+{\theta }_2{W}_n)=u(W_1,W_2,\dots ,W_n)$. So, the distribution of $Z$ does not depend on ${\theta }_1,{\theta }_2$. In this case, $Z$ is Ancillary Statistic to $\theta$, and it is called location- and scale-invariant statistic.