Chasing the peak: optimal statistics for weak shear analyses
  Weak gravitational lensing analyses are fundamentally limited by the
intrinsic, non-Gaussian distribution of galaxy shapes. We explore alternative
statistics for samples of ellipticity measurements that are unbiased,
efficient, and robust. We take the non-linear mapping of gravitational shear
and the effect of noise into account. We then discuss how the distribution of
individual galaxy shapes in the observed field of view can be modeled by
fitting Fourier modes to the shear pattern directly. We simulated samples of
galaxy ellipticities, using both theoretical distributions and real data for
ellipticities and noise. We determined the possible bias $\Delta e$, the
efficiency $\eta$ and the robustness of the least absolute deviations, the
biweight, and the convex hull peeling estimators, compared to the canonical
weighted mean. Using these statistics for regression, we have shown the
applicability of direct Fourier mode fitting. These estimators can be unbiased
in the absence of noise, and decrease noise bias by more than $\sim 30\%$. The
convex hull peeling estimator distribution is centered around the underlying
shear, and its bias least affected by noise. The least absolute deviations
estimator to be the most efficient estimator in almost all cases, except in the
Gaussian case, where it's still competitive ($0.83<\eta <5.1$) and therefore
robust. These results hold when fitting Fourier modes, where amplitudes of
variation in ellipticity are determined to the order of $10^{-3}$. The peak of
the ellipticity distribution is a direct tracer of the underlying shear and
unaffected by noise, and we have shown that estimators that are sensitive to a
central cusp perform more efficiently, potentially reducing uncertainties by
more than $50\%$ and significantly decreasing noise bias.
