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Thread: efficiency help

  1. #1
    Senior Member Danneedshelp's Avatar
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    efficiency help

    Q) $\displaystyle Y_{1},Y_{2},...,Y_{n}$ is a random sample from a normal distribution with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^{2}$. Two unbiased estimators of $\displaystyle \sigma^{2}$ are

    $\displaystyle \hat{\sigma_{1}}^{2}=S^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(Y_{i}-\bar{Y})^{2}$ and $\displaystyle \hat{\sigma_{2}}^{2}=\frac{1}{2}(Y_{1}-Y_{2})^{2}$.

    Find the efficiency of $\displaystyle \hat{\sigma_{1}}^{2}$ relative to $\displaystyle \hat{\sigma_{2}}^{2}$.

    I am stuck on finding the proper variance for $\displaystyle \hat{\sigma_{2}}^{2}$, $\displaystyle V(\hat{\sigma_{2}}^{2})$. My work indicates $\displaystyle V(\hat{\sigma_{2}}^{2})=\frac{\sigma^{4}}{2}$, because $\displaystyle \hat{\sigma_{2}}^{2}$ is simply $\displaystyle S^{2}$
    with $\displaystyle n=2$. Moreover, $\displaystyle V(\hat{\sigma_{1}}^{2})=\frac{2\sigma^{4}(n-1)}{n^{2}}$.

    So, the efficiency of $\displaystyle \hat{\sigma_{1}}^{2}$ relative to $\displaystyle \hat{\sigma_{2}}^{2}$ is $\displaystyle eff(\hat{\sigma_{1}}^{2},\hat{\sigma_{2}}^{2})=\fr ac{V(\hat{\sigma_{2}}^{2})}{V(\hat{\sigma_{1}}^{2} )}=\frac{n^{2}}{4(n-1)}$.

    The book's answer is $\displaystyle n-1$.

    I am stuck, so any help would be great.

    Thanks
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  2. #2
    MHF Contributor matheagle's Avatar
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    nope, use...

    $\displaystyle {(n-1)S^2\over \sigma^2}={\sum_{i=1}^n(X_i-\bar X)^2\over n-1}\sim \chi^2_{n-1}$

    Thus

    $\displaystyle V(S^2)=V\left[\left({(n-1)S^2\over \sigma^2}\right)\left({\sigma^2\over n-1}\right)\right]$

    $\displaystyle =\left({\sigma^4\over (n-1)^2}\right)V\left(\chi^2_{n-1}\right)$

    $\displaystyle =\left({\sigma^4\over (n-1)^2}\right)2(n-1)$

    $\displaystyle = {2\sigma^4\over n-1}$

    And, yes the other point estimator is a sample variance with n=2.

    So $\displaystyle V(\hat\sigma_2^2)= {2\sigma^4\over 2-1} =2\sigma^4$

    So, the ratio is n-1 or 1/(n-1) depending on which way you divide.
    Last edited by matheagle; Mar 1st 2010 at 10:51 PM.
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