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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?

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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
up vote 2 down vote accepted

Your calculation of MSE looks fine, and if this is homework and MSE was the criterion you were supposed to use, then I guess they are "equally" good.

However, I'd qualitatively note that the first estimator is less precise (higher variance) and more accurate (zero bias), while the second is more precise (lower variance) and less accurate (clear bias). So if this were a scientific measurement, there would be good reasons to prefer the first estimator because of its lower bias, despite the higher variance.

EDIT: TheJoker's point about no "best" estimate is correct without some other context. This is why I explicitly used the example of a scientific measurement. In that case, bias is more important than precision, because you can usually increase precision over time by doing additional experiments. Unless you know about the bias, this can be difficult to correct.

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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.

However (at leastin my opinion) there is no such thing as the best estimator. Definition of best varies according to the situation we are dealing with. Unbiasedness is very good and desired property (hence we are most concerned with UMVUE or BUEs) but it might not be best for all purposes.

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