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In weak gravitational lensing, weighted quadrupole moments of the brightnessprofile in galaxy images are a common way to estimate gravitational shear. Weemploy general adaptive moments (GLAM) to study causes of shear bias on afundamental level and for a practical definition of an image ellipticity. TheGLAM ellipticity has useful properties for any chosen weight profile: theweighted ellipticity is identical to that of isophotes of elliptical images,and in absence of noise and pixellation it is always an unbiased estimator ofreduced shear. We show that moment-based techniques, adaptive or unweighted,are similar to a model-based approach in the sense that they can be seen asimperfect fit of an elliptical profile to the image. Due to residuals in thefit, moment-based estimates of ellipticities are prone to underfitting biaswhen inferred from observed images. The estimation is fundamentally limitedmainly by pixellation which destroys information on the original, pre-seeingimage. We give an optimized estimator for the pre-seeing GLAM ellipticity andquantify its bias for noise-free images. To deal with pixel noise, we considera Bayesian approach where the posterior of the GLAM ellipticity can beinconsistent with the true ellipticity if we do not properly account for ourignorance about fit residuals. This underfitting bias is S/N-independent butchanges with the pre-seeing brightness profile and the correlation orheterogeneity of pixel noise over the post-seeing image. Furthermore, wheninferring a constant ellipticity or, more relevantly, constant shear from asource sample with a distribution of intrinsic properties (sizes, centroidpositions, intrinsic shapes), an additional, now noise-dependent bias arisestowards low S/N if incorrect priors for the intrinsic properties are used. Wediscuss the origin of this prior bias.

Weak-lensing shear estimates with general adaptive moments, and studies of bias by pixellation, PSF distortions, and noise