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Estimator / Statistics

  1. Aug 5, 2013 #1
    I'm a bit confused about a particular step in a calculation.

    Given Theta_n = (2/n)(X1 + X2 + ... + Xn) being an unbiased estimator of Theta for a U(0,Theta), we have to prove it by showing E(Theta_n) = Theta.

    And we go on E(Theta_n) = (2/n)E(X1+X2 + .. Xn)

    Now at this point the solution is (2/n) * n * (Theta/2) (= Theta which is the sought-after result)

    I understand that Theta/2 is the mean of a U() but how exactly does one go from E(X1 + X2 + .. Xn) to equaling it to n*E(Xi)? Is E(X1) = E(X2) = E(Xi)? If yes, why?

    (PS. A more complex example is Var(X1+X2 + .. Xn) appearing to also result to nV(Xi) (=nσ^2) )
     
  2. jcsd
  3. Aug 5, 2013 #2
    Hrm. Afterthought. I guess it might be simply a blatant case of "mean of the whole but it missed the 'over n' so it's n * E(Xi)".

    I guess it might apply in the case of Var too..
     
  4. Aug 5, 2013 #3
    I'm having a difficulty seeing how that could be true for Var.

    Var((2/n)(X1 + X2 + ... + Xn)) = (4/n^2)nVar(Xi).

    I understand 4/n^2 going out as a property of Var but how is Var(X1 + X2 + ... + Xn) = nVar(Xi)?
     
  5. Aug 5, 2013 #4
    Nevermind. I found it. If X1, X2.. are uncorrelated then V(Σ(Xi)) = ΣV(Xi) ..after a proof involving Var's equality with E[X^2) - E^2
     
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