# Choosing the better of two estimators

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?

• What about $d_3=d_2-2$? – Did Nov 11 '12 at 12:56
• @did In that case, $d_3$ would be the best of the three because the MSE equals 2. Is that correct? – woaini Nov 11 '12 at 13:02
• Obviously.   – Did Nov 11 '12 at 13:04

Since MSE is same for both estimators, you can check for some other properties of estimators. One very important property is Bias of estimator. Here, estimator $d_1$ is unbiased whereas estimator $d_2$ is not. So you can consider estimator $d_1$ to be better.