Measurement uncertainty limit analysis with the Cramér-Rao bound in case of biased estimators

• Relations between a biased and the bias-corrected estimator are investigated.
• Using the Cramér-Rao bound, the efficiency of both estimators is compared.
• As a result, the efficiency of both estimators is (at least asymptotically) equal.
• If the efficiency is one, the mean square error is also minimal.
• It is therefore sufficient to evaluate the biased or the bias-corrected estimator.

The Cramér-Rao lower bound has been proven to be a valuable tool for determining the minimal achievable measurement uncertainty and for analyzing the performance of estimators in terms of efficiency. While this is common for unbiased estimators, a bias does often occur in practice. The performance analysis of biased estimators is more difficult, because the bias has to be taken into account additionally. Furthermore, not the behavior of the biased estimator is finally of interest in measurements, but the behavior of its bias-corrected counterpart. In order to simplify the required performance analysis for biased estimators, the relation between the efficiencies of the biased and the bias-corrected estimator is derived. As result, both efficiencies are shown to be (at least asymptotically) identical. Hence, the bias-corrected estimator attains the Cramér-Rao bound if and only if the biased estimator attains its Cramér-Rao bound. Furthermore, the mean square errors become minimal if and only if the estimators reach the Cramér-Rao bound. Consequently, the optimality of the bias-corrected estimator can also be judged by evaluating the mean square error of the biased estimator.