I'm having some difficulty with unbiased estimators, and wondered if anyone could help me. I believe I understand the general concepts OK, however when I come to look at some sample questions to test my understanding, I feel a little lost! I must stress that this is not homework, or any other assessed work - purely for my own understanding, as statistic isn't an area I have studied in great detail to he truthful. For instance, one question I would like to get my head around is:
$X \sim B(n,p)$ and $P=bX$. Assuming $n$ is known, determine $b$ such that $P$ is an unbiased estimator of $p$ and find its standard error.
To start with, I have found the following:
$E(P)=E(bX)=bE(X)=bnp$ - therefore, am I right in thinking that for P to be an inbiased estimator, we must have b equal to $1/n$?
For the second part, I believe that I require the variance, as follows:
$V(P)=V(bX)=b^2V(X)=b^2 np(1-p)$, and so where $b=1/n$ we have:
$V(P)=\frac{p(1-p)}{n}$ and so the standard error will be:
Standard Error $=\sqrt{\frac{p(1-p)}{n}}$.
Does this solution look correct - I feel that it is, however would like to check my understanding.
Thanks in advance for any help!
Best, Chris
For an estimator $\hat{\theta}$ to be an unbiased estimator of $\theta$, it is required that $$ E(\hat{\theta})=\theta. $$ So for your example, using the estimator $\hat{p}=bX$ for $p$, you are indeed correct in your conclusion that for it to be unbiased $b=1/n$. As you showed, $$ E(\hat{p})=E(bX)=bE(X)=bnp\Longrightarrow E(\hat{p})=p \text{ if } b=\frac{1}{n}. $$
Your workings for the standard error are correct too. It seems like you've got things under control.