Variance of an unbiased estimator

• May 14th 2011, 11:06 PM
TheFirstOrder
Variance of an unbiased estimator
Hello guys! (Hi)

I am having problems with the following problem (from Cochran's "Sampling Techniques", 3rd edition, p.49, question 2.19):

Suppose in a population of N units, the value of ${Y}_{1}$ is very small while that of ${Y}_{n}$ is very large as compared to other values of the variable of interest. Show that for simple random sample of size n the following estimator is unbiased for estimating $\bar{Y}$.

${\hat{\bar{Y}}}_{S}$=

{ $\bar{y} +c$ if the sample contains ${Y}_{1}$ but not ${Y}_{n}$
{ $\bar{y} -c$ if the sample contains ${Y}_{n}$but not ${Y}_{1}$
{ $\bar{y}$ for all other samples

where c is a fixed constant.

Prove that:

$Var({\hat{\bar{Y}}}_{S}) = (1-\frac{n}{N} )(\frac{S^{2}}{n }-\frac{2c}{(N-1) }({Y}_{n}-{Y}_{1}-nc) )$

Okay, firstly, to show that the estimator is unbiased, is the following method correct?

Remembering that $E(\bar{y})=\bar{Y}$,

$Bias({\bar{Y}}_{S})=E(\bar{y} +c)-\bar{Y}=0$
$Bias({\bar{Y}}_{S})=E(\bar{y} -c)-\bar{Y}=0$
$Bias({\bar{Y}}_{S})=E(\bar{y})-\bar{Y}=0$

But then I get stuck on how to prove how to prove the variance of the estimator. I believe that the following formula for the variance of the sample mean from a simple random sample without replacement:

$V({{\bar{y}})=(\frac{1}{n})(1- \frac{n}{N})({S}^{2})$

might come in handy but I am really stuck as I do not even know where to begin to find this required variance. I am really stuck on this one and any hints to help me would be GREATLY appreciated.