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Thread: Mean quadratic error

  1. #1
    Junior Member
    Oct 2009

    Mean quadratic error

    To prove that mean quadratic error of a parameter is the sum of varianca of parameter and quadratic B of parameter. Thanks
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  2. #2
    MHF Contributor
    May 2010
    Did you mean estimator instead of parameter? Outside of bayesian statistics, parameters do not normally have a variance.

    Estimate: X
    True Value: Y

    $\displaystyle E(X-Y)^2 = E(X^2) - 2E(XY) + E(Y)^2$

    $\displaystyle E(X-Y)^2 = E(X^2) - 2YE(X) + Y^2$

    $\displaystyle E(X-Y)^2 = E(X^2) - 2YE(X) + Y^2 + E(X)^2 - E(X)^2$

    $\displaystyle E(X-Y)^2 = E(X^2) - E(X)^2 - 2YE(X) + Y^2 + E(X)^2$

    $\displaystyle E(X-Y)^2 = Var(X) + Y^2 - 2YE(X) + E(X)^2$

    You may find it interesting to note that the quatratic ("B") factorises to give

    $\displaystyle E(X-Y)^2 = Var(X) + E(Y - E(X))^2 $

    So that for an unbiased estimator, the MSE is equal to the variance of the estimate (as you'd expect from the definition of a variance).
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