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Math Help - Mean quadratic error

  1. #1
    Junior Member
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    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
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    Did you mean estimator instead of parameter? Outside of bayesian statistics, parameters do not normally have a variance.

    Estimate: X
    True Value: Y

    E(X-Y)^2 = E(X^2) - 2E(XY) +  E(Y)^2

    E(X-Y)^2 = E(X^2) - 2YE(X) +  Y^2

    E(X-Y)^2 = E(X^2) - 2YE(X) +  Y^2 + E(X)^2 - E(X)^2

    E(X-Y)^2 = E(X^2) - E(X)^2  - 2YE(X) +  Y^2  + E(X)^2

    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

    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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