$X_1, X_2\dots , X_n$ are independent Bernoulli r.v. s.t. $P(X_i=1)=p$, $i=1,2,\dots , n$, where $p\in [1/2,1)$.
(a) Is $T=\frac{1}{n} \sum_{i=1}^{n}X_i$ MVUE of $p$? Justify it.
It can be easily shown that $T$ is MVUE if $p\in (0,1)$, since $T$ is complete sufficient statistics(as belongs to exponential family) and unbiased estimator by Lehmann-Scheffe theorem. But here $p\in(1/2,1]$. I don't think the result would be same(or may be same). I am confused!!
What I did is that:
Let $T^*=a\space \text{if $T<.5$ and}\\ \space\space\space\space\space\space\space\space bT\space \text{if $T\geq.5$}$
$a,b$ are such that $T^*$ is u.e. of $p.$
Then I tried to find $a,b$ such that $Var(T^*)<Var(T)$. But I don't think this is right.