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As modern-day precision cosmology aims for statistical uncertainties of thepercent level or lower, it becomes increasingly important to reconsiderestimator assumptions at each step of the process, and their consequences onthe statistical variability of the scientific results.  We compare $L^1$ regression statistics to the weighted mean, the canonical$L^2$ method based on Gaussian assumptions, for inference of the weakgravitational shear signal from a catalog of background ellipticitymeasurements around a sample of clusters, in many recent analyses a standardstep in the process.  We use the shape measurements of background sources around 6925 AMICOclusters detected in the KiDS 3rd data release. We investigate the robustnessof our results and the dependence of uncertainties on the signal-to-noiseratios of the background source detections. Using a halo model approach, wederive lensing masses from the estimated excess surface density profiles.  The highly significant shear signal allows us to study the scaling relationbetween the $r$-band cluster luminosity $L_{200}$, and the derived lensing mass$M_{200}$. We show the results of the scaling relations derived in 13 bins in$L_{200}$, with a tightly constrained power law slope of $\sim 1.24\pm 0.08$.We observe a small, but significant relative bias of a few percent in therecovered excess surface density profiles between the two regression methods,which translates to a $1\sigma$ difference in $M_{200}$. The efficiency of$L^1$ is at least that of the weighted mean, relatively increasing with highersignal-to-noise shape measurements.  Our results indicate the relevance of optimizing the estimator for inferingthe gravitational shear from a distribution of background ellipticities. Theinterpretation of measured relative biases can be gauged by deeperobservations, while increased computation times remain feasible.

AMICO galaxy clusters in KiDS-DR3: the impact of estimator statistics on the luminosity-mass scaling relation