  Context. The density split statistics in weak gravitational lensing analyses
probes the correlation between regions of different (foreground) galaxy number
densities and their weak lensing signal, measured by the shape distortion of
background galaxies. Aims. In this paper, we reconsider density split
statistics, by constructing a new angular filter function that is adapted to
the expected relation between galaxy number density and shear pattern, in a way
that the filter weighting the galaxy number density is matched to the filter
that is used to quantify the shear signal. Methods. We use the results of
numerical ray-tracing simulations, specifically through Millennium Simulation
supplemented by a galaxy distribution based on a semi-analytic model, to
construct a matched pair of adapted filter functions for the galaxy density and
the tangential shear signal. We compare the performance of our new filter to
the previously used top-hat filter, applying both to a different and
independent set of numerical simulations (SLICS, cosmo-SLICS). Results. We show
that the adapted filter yields a better correlation between the total matter
and the galaxy distribution. Furthermore, the adapted filter provides a larger
signal-to-noise ratio to constrain the bias between the total matter and the
galaxy distribution, and we show that it is, in general, a more sensitive
discriminator between different cosmologies, with the exception of cosmologies
with very large $\sigma_8$ values. All analyses lead to the conclusion that our
adapted filter should be favored in future density split statistic works.
