Unbiased estimators

An estimator \(T = g(X_1, \dots, X_n)\) for parameter \(\theta\) is called unbiased if:

\[E_\theta(T) = \theta\]

where \(E_\theta(T)\) denotes the expected value of T under the assumption that the true parameter value is \(\theta\)
Thus: If we determine the expected value of the estimation statistic T under the assumption that the true parameter value is \(\theta\), we get \(\theta\) as the expected value. Value of \(\theta\) is neither systematically overestimated nor underestimated

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The bias of an estimator $T = g(X_1, , X_n) for parameter \(\theta\) is given by

\[Bias_\theta(T) = E_\theta(T) - \theta\]

An estimator / estimation statistic T for a parameter \(\theta\) is unbiased if and only if \(Bias_\theta(T) = 0\) holds

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The standard error of an unbiased estimator \(T = g(X_1, \dots, X_n)\) is given by its standard deviation:

\[o_T = \sqrt{Var_\theta(T)}\]

The standard error depends on the (unknown) distribution of X and needs to be estimated itself

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