the bias correction function $\symBiasCorrectionFunc$ that satisfies $\symCardinality = \symExpectation(\symBiasCorrectionFunc(\symCardinalityRawEstimate)\vert\symCardinality)$ for all cardinalities. By definition, the estimator $\symCardinalityRawEstimate':=\symBiasCorrectionFunc(\symCardinalityRawEstimate)$ is unbiased and a function of the raw estimator. Its standard deviation is compared with that of the improved estimator in \[fig:stdev\_comparison\]. For cardinalities smaller than the empirical bias correction approach is not very precise. This is the reason why all previous approaches had to switch over to the linear counting estimator at some point. The standard deviation of the linear counting estimator is also shown in \[fig:stdev\_comparison\]. Obviously, the previous approaches cannot do better than given by the minimum of both curves for linear counting and raw estimator. In practice, the standard deviation of the combined approach is even larger, because the choice between both estimators must be made based on an estimate and not on the true cardinality, for which the intersection point of both curves, which is approximately in \[fig:stdev\_comparison\], would be the ideal transition point. In contrast, the improved estimator performs well over the entire cardinality range.

We also investigated the estimation error of the improved estimator for the case $\symPrecision=22$, $\symRegRange=10$ as shown in \[fig:raw\_corrected\_estimation\_error\_22\_10\]. The standard deviation is by a factor 32 smaller according to the 