Is there a case when uniform distribution posses unbiased and umvue estimate of $\theta$?
Suppose
$X_1, X_2, \dots, X_{45}$ ~ uniform on interval $[\theta-1/2\ , \theta+1/2]$
My views: I know $\max(X_1, X_2, \dots, X_{45})$ and $\min(X_1, X_2, \dots, X_{45})$ are mle.
2 sample mean is unbiased in case when lower limit is $0$ for uniform distribution.
What gave me doubt is.. this is a symmetric distribution, so what property does sample mean and median has for this distribution over this limits?
Edit : Adding more details, My problem is I want to know if sample mean and sample median are unbiased for this particular distribution.? My intuition says yes, but I couldn’t connect it with the theory.
Actually the likelihood is
$$L(\theta)=\mathbb{1}_{[x_{(n)}-0.5;x_{(1)}0.5]}(\theta)$$
Thus any value in the interval is MLE. As an example you can take the midrange
$$T_1=\frac{x_{(1)}+x_{(n)}}{2}$$
as a MLE.
Also $T_2=(X_{(1)},X_{(n)})$ is sufficient, minimal but the problem is that this estimator is NOT complete.
In fact the given density belongs to the Location Family and $(x_{(n)}-x_{(1)})$ is location invariant too, thus it is also ancillary and thus
$$\mathbb{E}[X_{(n)}-X_{(1)}]=C$$
where C does not depend on $\theta$ which implies that, $\forall \theta$
$$\mathbb{E}[X_{(n)}-X_{(1)}-C]=0$$
and clearly, $\forall \theta$
$$\mathbb{P}[X_{(n)}-X_{(1)}-C=0] \neq 1$$
With these preliminary statements, perhaps it is better that you add all the details of your problem.