Scale-Invariant Statistic

Definition

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