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Thread: Show that s^2 is an unbiased estimator for sigma^2

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    Show that s^2 is an unbiased estimator for sigma^2




    I am trying to complete a proof showing that s^2 = sum(Yi - Yi hat)^2/(n-2) is unbiased for sigma^2

    In other words, show that E[s^2] = sigma^2

    I have...

    E[s^2] = E [sum(yi - yi hat)^2/(n-2)]

    which is equivalent to...

    E[s^2(n-2)] = E[sum(yi - yi hat)^2]

    From here,

    E[sum(yi - yi hat)^2] = Var(Yi - Yi hat) + [E(Yi - Yi hat)]^2

    So now I have to show that Var(Yi - Yi hat) + [E(Yi - Yi hat)]^2 = sigma^2(n-2), but I keep getting stuck at this point, especially in terms of taking the variance...should I make a substitution for yi hat? Anything I try doesn't seem to get me where I want!

    If anyone can provide me with some direction that would be great, thanks

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    Quote Originally Posted by statgirl11 View Post

    If anyone can provide me with some direction that would be great, thanks

    Someone else asked this question a few days ago. See my reply here:

    http://www.talkstats.com/showthread.php?t=9685

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    I looked at your solution, however you have...

    Subtracting (2) from (1) gives
    (3) (Y_i Ybar) = Beta1(X_i Xbar) + (u_i ubar), and go from there...

    however, there is a difference in that you are using Y_i - Ybar, rather than Y_1 - Yhat, which is what I am trying to do...

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    Quote Originally Posted by statgirl11 View Post
    I looked at your solution, however you have...


    however, there is a difference in that you are using Y_i - Ybar, rather than Y_1 - Yhat, which is what I am trying to do...
    I think you mean: Y_i - Yhat_i (?).

    This is what e_i is in (4) and (5)...i.e. e_i = Y_i - Yhat_i

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    Yes that is what I meant, and I see what you mean now, thank you!

    I'm going to try and work through your other post and see if I can figure it out from there.

    Thanks a lot

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