3. - Consider a finite size population for N in which you want to study a variable y that takes a very small value for the first element of the sampling frame y1 and a very large value for the last element of the sampling frame yN. In order to estimate the average ? of this population µ, it is proposed, based on a Simple Random Sampling without replacement of size n, to have an estimator of the form:
?c is the estimator which has 3 parts:
?c = ? + c , if y1 belongs to the sample and yN does not belong to the sample
?c = ? - c , if y1 does not belong to the sample and yN belongs to the sample
?c = ? ,And, in another case,
,where c is a positive constant.
a) Is ?c an unbiased estimator of µ?
b) Find the variance of ?c.
c) Are there values for c that make ?c to have less variance than ?? Does this contradict that Y is the best unbiased estimator of µ?
3. - Consider a finite size population for N in which you want to study a variable y that takes a very small value for the first element of the sampling frame y1 and a very large value for the last element of the sampling frame yN. In order to estimate the average Ȳ of this population μ, it is proposed, based on a Simple Random Sampling without replacement of size n, to have an estimator of the form: Ȳc is the estimator which has 3 parts: Ȳc = Ȳ + c , if y1 belongs to the sample and yN does not belong to the sample Ȳc = Ȳ - c , if y1 does not belong to the sample and yN belongs to the sample Ȳc = Ȳ ,And, in another case, ,where c is a positive constant. a) Is Ȳc an unbiased estimator of μ? b) Find the variance of Ȳc. c) Are there values for c that make Ȳc to have less variance than Ȳ? Does this contradict that Y is the best unbiased estimator of μ?