Is statistic $\max\{-X_{(1)},X_{(n)}\}$ one dimension or two dimension?
I was trying to find the minimal sufficient statistic for $U(-\theta,\theta)$ from $n$ $i.i.d$ random variables $X_i$. The result is that $\theta\ge \max\{-X_{(1)},X_{(n)}\}$ thus the minimal sufficient statistic is $\max\{-X_{(1)},X_{(n)}\}$. However, the problem actually states that "Find a two dimensional minimal sufficient statistic for $U(-\theta,\theta)$". Is $\max\{-X_{(1)},X_{(n)}\}$ a two dimensional statistic?
Here $X_{(i)}$ is the $i^{th}$ smallest value of $X_1,\cdots,X_n$.
I presume by $\{X_{(i)}\}$ you mean the ordered sequence constructed from $\{X_i\}$. The minimal sufficient statistics is the pair of $(min\{X_i\}, max\{X_i\})$, however, if the sequence is ordered, then you are right; the minimal sufficient statistics is $max\{-X_{(1)}, X_{(n)}\}$, but it requires an additional step of ordering.
Correction
Yes. You are right. $max\{|X_i|\}$ is the sufficient statistics.