Solution
= Solution
Since $E X_{(1)}=\theta+(1-\theta)/(n+1)$, take $\hat\theta=((n+1)X_{(1)}-1)/n$. It is unbiased, and the supplied variance formula gives $\operatorname{var}\hat\theta=(1-\theta)^2/[n(n+2)]$, which is its MSE.
= Solution
Since $E X_{(1)}=\theta+(1-\theta)/(n+1)$, take $\hat\theta=((n+1)X_{(1)}-1)/n$. It is unbiased, and the supplied variance formula gives $\operatorname{var}\hat\theta=(1-\theta)^2/[n(n+2)]$, which is its MSE.