Hello guys!
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 is very small while that of 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 .
=
{ if the sample contains but not
{ if the sample contains but not
{ for all other samples
where c is a fixed constant.
Prove that:
Okay, firstly, to show that the estimator is unbiased, is the following method correct?
Remembering that ,
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:
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.