Results 1 to 2 of 2

Math Help - efficiency help

  1. #1
    Senior Member Danneedshelp's Avatar
    Joined
    Apr 2009
    Posts
    303

    efficiency help

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

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

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

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

    So, the efficiency of \hat{\sigma_{1}}^{2} relative to \hat{\sigma_{2}}^{2} is 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 n-1.

    I am stuck, so any help would be great.

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    nope, use...

    {(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

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

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

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

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

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

    So 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; March 1st 2010 at 10:51 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Efficiency of p in binom(n,p)
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: September 15th 2010, 06:18 PM
  2. Efficiency Problems
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: February 12th 2010, 06:20 AM
  3. Work Efficiency
    Posted in the Calculus Forum
    Replies: 5
    Last Post: June 5th 2009, 06:54 AM
  4. Relative efficiency
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: December 7th 2008, 12:10 AM
  5. Efficiency
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 2nd 2008, 11:18 PM

Search Tags


/mathhelpforum @mathhelpforum