  The traditional Schmidt density estimator has been proven to be unbiased and
effective in a magnitude-limited sample. Previously, efforts have been made to
generalize it for populations with non-uniform density and proper
motion-limited cases. This work shows that the then-good assumptions for a
proper motion-limited sample are no longer sufficient to cope with modern data.
Populations with larger differences in the kinematics as compared to the local
standard of rest are most severely affected. We show that this systematic bias
can be removed by treating the discovery fraction inseparable from the
generalized maximum volume integrand. The treatment can be applied to any
proper motion-limited sample with good knowledge of the kinematics. This work
demonstrates the method through application to a mock catalogue of a white
dwarf-only solar neighbourhood for various scenarios and compared against the
traditional treatment using a survey with Pan-STARRS-like characteristics.
