Validation of Selection Function, Sample Contamination and Mass
  Calibration in Galaxy Cluster Samples
  We construct and validate the selection function of the MARD-Y3 sample. This
sample was selected through optical follow-up of the 2nd ROSAT faint source
catalog (2RXS) with Dark Energy Survey year 3 (DES-Y3) data. The selection
function is modeled by combining an empirically constructed X-ray selection
function with an incompleteness model for the optical follow-up. We validate
the joint selection function by testing the consistency of the constraints on
the X-ray flux--mass and richness--mass scaling relation parameters derived
from different sources of mass information: (1) cross-calibration using SPT-SZ
clusters, (2) calibration using number counts in X-ray, in optical and in both
X-ray and optical while marginalizing over cosmological parameters, and (3)
other published analyses. We find that the constraints on the scaling relation
from the number counts and SPT-SZ cross-calibration agree, indicating that our
modeling of the selection function is adequate. Furthermore, we apply a largely
cosmology independent method to validate selection functions via the
computation of the probability of finding each cluster in the SPT-SZ sample in
the MARD-Y3 sample and vice-versa. This test reveals no clear evidence for
MARD-Y3 contamination, SPT-SZ incompleteness or outlier fraction. Finally, we
discuss the prospects of the techniques presented here to limit systematic
selection effects in future cluster cosmological studies.
