Find a two dimensional minimal sufficient statistic for $\theta$ from $n$ independent random variables $X_k\sim > U(-k\theta+k,k\theta+k)$, $k\in\{1,\cdots,n\}$
Here is what I've attempted.
The pdf of X is $$\delta_\theta(x)=\prod_{i=1}^n\frac{1}{2k\theta}\mathbb{1}_{(1-\theta)k\leq x_k\leq(1+\theta)k}.$$ This means that $$\theta\ge\max\{1-\frac{x_k}{k},\frac{x_k}{k}-1\},\quad\forall k\in\{1,\cdots,n\}$$ or equivalently $|\frac{x_k}{k}-1|\leq\theta,\quad\forall k\in\{1,\cdots,n\}$. What I've got is $\underset{k\in\{1,\cdots,n\}}{\max}\left\{|\frac{x_k}{k}-1|\right\}$ as a minimal sufficient statistic. However, I am not sure if this is correct and the question was about there should be a 2-dimensional minimal statistic.
--Update It is always possible that the problem itself is wrong, i.e. there is no 2-dim minimal sufficient statistic. If this is the case, how to disprove it?