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Math Help - What is the Varinace of the sample variance of iid normal distributed variables

  1. #1
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    What is the Varinace of the sample variance of iid normal distributed variables

    Hi Guys
    I have a sample of N iid normal distributed variables. And I need to calculate the Variance of the Maximumlikelihood estimated variance (sample variance)  {\sum (X_i-\bar X)^2\over N}

    Can somebody help me. I just can't come to an answer

    Cheers
    Todor
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  2. #2
    Moo
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    Hello,

    This rv follows a \chi^2(N-1) distribution So if you look at a table you should get its variance
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  3. #3
    MHF Contributor matheagle's Avatar
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    Nope, the sum over \sigma^2 has a chi-square distribution.


     {\sum_{i=1}^n(X_i-\bar X)^2\over n}= \left({\sigma^2\over n}\right)\left({\sum_{i=1}^n(X_i-\bar X)^2\over \sigma^2}\right)

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

    Hence  V\left( {\sum_{i=1}^n(X_i-\bar X)^2\over n}\right)= V\left( {\sigma^2\over n}\chi^2_{n-1}\right)

    =\left( {\sigma^4\over n^2}\right)2(n-1)
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  4. #4
    Moo
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    Quote Originally Posted by matheagle View Post
    Nope, the sum over \sigma^2 has a chi-square distribution.
    I wonder where I wrote that it doesn't
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