Completeness II: A signal-to-noise approach for completeness estimators
  applied to galaxy magnitude-redshift surveys
  This is the second paper in our completeness series which addresses some of
the issues raised in the previous article by Johnston, Teodoro & Hendry (2007)
in which we developed statistical tests for assessing the completeness in
apparent magnitude of magnitude-redshift surveys defined by two flux limits.
The statistics, Tc and Tv, associated with these tests are non-parametric and
defined in terms of the observed cumulative distribution function of sources;
they represent powerful tools for identifying the true flux limit and/or
characterising systematic errors in magnitude-redshift data. In this paper we
present a new approach to constructing these estimators that resembles an
"adaptive smoothing" procedure - i.e. by seeking to maintain the same amount
the information, as measured by the signal-to-noise ratio, allocated to each
galaxy. For consistency with our previous work, we apply our improved
estimators to the Millennium Galaxy Catalogue (MGC) and the Two Degree Field
Galaxy Redshift Survey (2dFGRS) data, and demonstrate that one needs to use a
signal-to-noise appropriately tailored for each individual catalogue to
optimise the performance of the completeness estimators. Furthermore, unless
such an adaptive procedure is employed, the assessment of completeness may
result in a spurious outcome if one uses other estimators present in the
literature which have not been designed taking into account "shot noise" due to
sampling.
