Let Y1<Y2<...<Yn be the order statistics of a random sample from a distribution with pdf $\displaystyle f(x; \theta) = 1$, $\displaystyle \theta - 0.5 < x < \theta + 0.5$. Show that every statistic u(X1,X2,...,Xn) such that $\displaystyle Y_n - 0.5<u(X_1,X_2,...,X_n)<Y_1 + 0.5$ is a mle of theta. In particular $\displaystyle (4Y_1 + 2Y_n + 1)/6, (Y_1 + Y_n)/2, (2Y_1 + 4Y_n - 1)/6$ are three such statistics.

I don't even know how to start this problem. Can someone please point me in the right direction?