Completeness I : revisited , reviewed and revived

We have extended and improved the statistical test recently developed by Rauzy for assessing the completeness in apparent magnitude of magnitude-redshift surveys. Our improved test statistic retains the robust properties – specifically independence of the spatial distribution of galaxies within a survey – of the Tc statistic introduced in Rauzy’s seminal paper, but now accounts for the presence of both a faint and bright apparent magnitude limit. We demonstrate that a failure to include a bright magnitude limit can significantly affect the performance of Rauzy’s Tc statistic. Moreover, we have also introduced a new test statistic, Tv, defined in terms of the cumulative distance distribution of galaxies within a redshift survey. These test statistics represent powerful tools for identifying and characterising systematic errors in magnitude-redshift data. We discuss the advantages of the Tc and Tv statistics over standard completeness tests, particularly the widely used V/Vmax test which assumes spatial homogeneity, and we demonstrate how our Tv statistic can essentially be regarded as an improved, cumulative V/Vmax test which makes better use of the magnitude completeness information in a redshift survey. Finally we apply our completeness test to three major redshift surveys: The Millennium Galaxy Catalogue (MGC), The Two Degree Field Galaxy Redshift Survey (2dFGRS), and the Sloan Digital Sky Survey (SDSS). We confirm that MGC and SDSS are complete up to the published (faint) apparent magnitude limit of mbj = 20.00 mag. and mr = 17.45 mag. respectively, indicating there are no residual systematic effects within the photometry. Furthermore, we show that, unless a bright limit is included for 2dFGRS, the data-set displays significant incompleteness at magnitudes brighter than the published limit of mbj = 19.45 mag.