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Math Help - Complete Statistic

  1. #1
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    Complete Statistic

    Let X_1,\cdots,X_n be a random sample from N(\mu,\mu^2). Show that T=(\sum_{i=1}^n X_i,\sum_{i=1}^n {X_i}^2) \mbox{is minimum sufficient but not complete}.

    so, I did the first part to show that T is a minimum sufficient statistic. But I am confused on how to show that it is not complete. How can I find a non zero function using t that does not depend on mu? I am confused on the expected value part. Could anyone help?
    Thanks.
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  2. #2
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    Quote Originally Posted by chutiya View Post
    Let X_1,\cdots,X_n be a random sample from N(\mu,\mu^2). Show that T=(\sum_{i=1}^n X_i,\sum_{i=1}^n {X_i}^2) \mbox{is minimum sufficient but not complete}.

    so, I did the first part to show that T is a minimum sufficient statistic. But I am confused on how to show that it is not complete. How can I find a non zero function using t that does not depend on mu? I am confused on the expected value part. Could anyone help?
    Thanks.
    Hint: It looks like \sum X_i and  \sum X_i ^ 2 can be used to estimate the same thing. Use this to get an unbiased estimate of 0.
    Last edited by theodds; February 13th 2011 at 02:46 PM.
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  3. #3
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    Ok. so \sum_{i=1}^n X_i \sim N(n\mu, n\mu^2) and E[(\sum_{i=1}^n X_i)^2]=n\mu^2+n^2\mu^2

    then find E_{\mu}[\sum_{i=1}^n {X_i}^2-(\sum_{i=1}^n {X_i})^2] = 0

    is this right or am I lost?
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  4. #4
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    Quote Originally Posted by chutiya View Post
    Ok. so \sum_{i=1}^n X_i \sim N(n\mu, n\mu^2) and E[(\sum_{i=1}^n X_i)^2]=n\mu^2+n^2\mu^2

    then find E_{\mu}[\sum_{i=1}^n {X_i}^2-(\sum_{i=1}^n {X_i})^2] = 0

    is this right or am I lost?
    You have the right idea, but you are going to need to scale one of those statistics to make the expectation 0 (they don't have the same expectation right off the bat, but the expectations are proportional).
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  5. #5
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    so if i do this:

    E_{\mu}[(n+1)\sum_{i=1}^n {X_i}^2-(\sum_{i=1}^n {X_i})^2] = [(n+1)\times n\mu^2] - [n\mu^2+n^2\mu^2]=0

    does this work?
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  6. #6
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    E_\mu \sum X_i ^ 2 = \sum E_\mu X_i ^ 2 = \sum (\mu^2 + \mu^2)

     E_\mu \left(\sum X_i \right)^2 = n\mu ^ 2 + n^2 \mu^2
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