$X_{1}, X_{2},...X_{n},n\geqslant 2, $ is a random sample from unif[$\theta -1, \theta +1$]
Followed with the problem, I got T(X)=($X_{(1)}, X_{(n)} $) is sufficient but not complete, But I got stuck in creating an unbiased estimator of $\theta$ that is a linear combination of $X_{(1)}$ and $ X_{(n)}$. My first thought was to find $EX_{(1)}$ and $ EX_{(n)}$, then manipulate somehow to have $aEX_{(1)}+bEX_{(n)}=\theta$.
However, when I started to find the expectation for each, I found it very tedious and wondered if there's other way to solve this problem. Or, is the sufficiency and incompleteness a hint for solving this problem? I haven't seen similar example and exercise on the book or online though...
Thanks for your help.
The natural linear combination is $\hat\theta=\frac12X_{(1)}+\frac12 X_{(n)}$
as a symmetry argument $E[X_{(1)}-\theta] =-E[X_{(n)}-\theta]$ will give $E[\hat\theta]=\theta$.