# $θ_2$ is better than $θ_1$ to estimate $μ$?

We offer two estimators for the average concentration $$μ$$ of lead in the atmosphere of a region of Quebec where factories manufacturing dyes are located. The first estimator $$θ_1$$ has a bias equal to $$0.2$$ and a variance of $$0.02$$. The second estimator $$θ_2$$ is unbiased and has a variance equal to $$0.06$$.

Which one is the best estimator?

I think $$θ_2$$ is better than $$θ_1$$ to estimate $$μ$$, but I am not sure.

EDIT

A PhD student in statistics explained to me that if $$MSE(\theta_1) = MSE(\theta_2)$$, then we cannot conclude. In other words, $$\theta_2$$ is not preferred over $$\theta_1$$ or inversely. I am not sure about that.

• I suppose these estimators have been chosen to have equal mean-squared error on purpose? Commented Nov 21, 2020 at 16:06
• @preferred_anon Yes, but the purpose is not well oriented because it causes confusion. Commented Nov 26, 2020 at 1:17

In classical statistics, an estimator is better than another if its MSE is lower.

In this case

$$MSE(\theta_1)=0.02+0.2^2=0.06$$

$$MSE(\theta_2)=\mathbb{V}[\theta_2]=0.06$$

Being

$$MSE(\theta_1)=MSE(\theta_2)$$

$$\theta_2$$ is preferred as it is unbiased

• Can you precise you answer? A PhD student in statistics explained to me that if $MSE(\theta_1) = MSE(\theta_2)$, then we cannot conclude. In other words, $\theta_2$ is not preferred over $\theta_1$ or inversely. I am not sure about that. Commented Nov 22, 2020 at 16:11
• @David: Given the fact that $MSE(\theta_1)=MSE(\theta_2)$ I do prefer the unbiased estimator VS the biased one..You and yuor friend do not? Commented Nov 22, 2020 at 16:18
• In its context, if you could give only one example where $\theta_2$ is preferred to $\theta_1$, then I will accept your answer and give you the bounty Commented Nov 23, 2020 at 17:54

In my experience, a situation like this can end up depending on the context, i.e. what you are trying to estimate. Since you are discussing the average concentration of lead in the atmosphere of a particular region, the results of this could be severe, in which case I have been taught taking the unbiased estimator is a better idea here (since your variance is still quite small).

Hope this helps!