Consider two estimators, $d_1$ and $d_2$ of a parameter $\theta$. If $E(d_1)=\theta$, $Var(d_1)=6$ and $E(d_2)=\theta+2$, $Var(d_2)=2$, which estimator is preferred?
What I did: The estimator with the lower MSE is better. $$E((d_1-\theta)^2) = Var(d_1)+(E(d_1)-\theta)^2 = 6+(\theta-\theta)^2 = 6 \\ E((d_2-\theta)^2) = Var(d_2)+(E(d_2)-\theta)^2 = 2+(\theta+2-\theta)^2 = 2+4=6$$ This means that both estimators are just as good.
Did i do this correctly?