Help with estimators Uni Economics

Could anyone help with Q2? Vaguely understand estimators but been trying to do this question for hours with no success. Help with Q3 and 4 would also be appreciated haha

thanks

Let $\hat{\mu}$ be one of the estimators you are given. Then $\hat{\mu}$ is an unbiased estimator for $\mu_X$ if and only if $E(\hat{\mu}) = \mu_X$.

I suggest you try them in the order (iii), (ii), (i). I'll show you (iii) and then see if you can continue.

$E(\hat{\mu}_3) = E((1/n) \sum_{1}^{n} X_{i}) = (1/n) E(\sum_{1}^{n} X_{i}) = (1/n) n E(X) = E(X) = \mu_X$

The second last step is due to the fact that the $X_i$ are iid. So this estimator is unbiased. Now see if you can do similar for (ii). Remember that the bias of the estimator is the difference between its expected value and $\mu_X$.

Part (i) is slightly trickier, but think about

$(1/n) \sum_{1}^{n} (X_i - \mu_X + \mu_X)^2)$

and connect one of the terms in its expansion with $\sigma^{2}_{X}$

Thanks so much for your help, for part ii) I've got n/n-2(E(X)) - Is that correct and if so is that what this biased is equal to?

For part i) i'm still quite confused about how you expanded it and how to link in $\sigma^{2}_{X}$ Sorry to be a pain

If you mean $E(\hat{\mu}_2) = \frac{n}{n-2}E(X) = \frac{n}{n-2} \mu_{X}$ then you are correct. For the bias, simply subtract off $\mu_X$ from this.



Here are two hints:

(i) $(x - a + a)^2 = (x-a)^2 + a^2 + 2 a (x-a)$

(ii) What is $(1/n) \sum_{1}^{n} (X_i - \mu_X)^2$ ? You should recognize it.

Thank you I now understand that, but how do I get from $(1/n) \sum_{1}^{n} (X_i)^2$ to $(1/n) \sum_{1}^{n} (X_i - \mu_X + \mu_X)^2)$ Am I missing some sort of rule/formula?

could I have some help with question 3 as well please?