that the function is positive for the chosen starting points. Even though the secant method has the disadvantage of slower convergence, a single iteration is simpler to calculate as it does not require the evaluation of the first derivative. Possible starting points are zero and the lower bound in . The iteration can be stopped, if the relative increment is below a certain limit $\symStopDelta$. Since the expected estimation error scales according to $1/\sqrt{\symNumReg}$, it makes sense to choose $\symStopDelta = \symStopEpsilon/\sqrt{\symNumReg}$ with some constant $\symStopEpsilon$. For our results presented below we used $\symStopEpsilon = 10^{-2}$. In practice, only a handful of iterations are necessary to satisfy the stop criterion.

Results {#sec:maximum_likelihood_estimation_error}
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![Relative error of the estimator for $\symPrecision = 12$ and $\symRegRange=20$.[]{data-label="fig:max_likelihood_estimation_error_12_20"}](max_likelihood_estimate_12_20){width="1\columnwidth"}

![Relative error of the estimator for $\symPrecision = 22$ and $\symRegRange=10$.[]{data-label="fig:max_likelihood_estimation_error_22_10"}](max_likelihood_estimate_22_10){width="1\columnwidth"}

We have investigated the estimation error of the estimator for both configurations as for the improved estimator. \[fig:max\_likelihood\_estimation\_error\_12\_20,fig:max\_likelihood\_estimation\_error\_22\_10\] look very similar to \[fig:raw\_corrected\_estimation\_error\_12\_20,fig:raw\_corrected\_estimation\_error\_22\_10\], respectively. For $\symPrecision=12$, $\symRegRange=20$ the estimator has a somewhat smaller median bias for cardinalities around . In addition, the standard deviation of the relative error is marginally better than that of the improved estimator for cardinalities between and as shown in \[fig:stdev\_comparison\]. 