Mean squared error and consistency

The (expected) mean squared error incorporates both bias and variance. The (expected) mean squared error of an estimator \(T = g(X_1, \dots, X_n)\) for the parameter \(\theta\) is given by

\[MSE_\theta(T) = E_\theta([T - \theta]^2)\]

The mean squared error indicates the expected deviation between the estimated T and the true parameter value \(\theta\)

• $MSE_(T) = Var_(T) + Bias_(T)^2
• if T is unbiased then \(MSE_\theta(T) = Var_\theta(T)\)

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An estimator \(T = g(X_1, \dots, X_n)\) for the parameter \(\theta\) is called consistent (in mean square), if

\[MSE_\theta(T) = Var_\theta(T) + Bias_\theta(T)^2 -> 0 for n -> \infty\]

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