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Thread: Variance of an unbiased estimator

  1. #1
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    Exclamation Variance of an unbiased estimator

    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 $\displaystyle {Y}_{1}$ is very small while that of $\displaystyle {Y}_{n}$ 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 $\displaystyle \bar{Y} $.

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

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

    where c is a fixed constant.

    Prove that:

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

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

    $\displaystyle 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.
    Last edited by TheFirstOrder; May 15th 2011 at 06:36 PM.
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