I see there were a few questions on SE about MLE of Uniform already but none of them helped me with this one:
We are to compute MLE of $U(\theta, \theta +1)$ and check if it is biased and consistent.
I tried by making a spin-off of an example with $U(0, \theta)$ but I am not sure if it is correct. Suppose there's $X_1, X_2, \dots, X_n$ i.i.d with $U(\theta, \theta +1)$, $T(X_1, \dots, X_n)$ is the statistic and $(x_1, \dots, x_n)$ a sample from that statistic.
I start of with computing $L(\theta)$
$$ L(\theta)=\prod_{i=1}^n\mathbb{1}_{[\theta, \theta +1]}(x_i) = \mathbb{1}_{(-\infty, X(1)]}(\theta)\cdot\mathbb{1}_{[X(n),\infty)}(\theta+1) $$ Since $P(x_i \geq \theta) = 1$ this is just $$ L(\theta)=\mathbb{1}_{[X(n),\infty)}(\theta+1) = \begin{cases} 1, & \text{if}\ \theta + 1 \geq X(n) \\ 0, & \text{otherwise} \end{cases} $$ The smallest value of $\theta = 1$ is then $\frac{X(n) - 1 + X(1)}{2}$ and this is our MLE. As @StubbornAtom pointed out in comments, this is not the only MLE possible.
How can I calculate bias and consistency of the $\hat{\theta}^{MLE}$ of my choosing?