Extracting the Possible Intrinsic Relation between Radiative Efficiency
  and Mass of QSOs: a Maximum Likelihood Method and its Application to the SDSS
  DR7 QSOs
  Radiative efficiencies of QSOs and its distribution encode rich information
on the evolution of both masses and spins of massive black holes (MBHs) across
cosmic time. In this paper, we develop a maximum likelihood method to
statistically extract the intrinsic relation between radiative efficiency
($\epsilon$) and mass ($M_{\bullet}$) of QSOs from their distribution on the
luminosity-(empirically estimated virial) mass plane. By using mock samples, we
find that strong constraint can be put on the $\epsilon-M_{\bullet}$ relation
at redshift $z\lesssim 0.4$ from uniform QSO samples similar to those in Sloan
Digital Sky Survey, and from QSO samples at $z \sim 0.6$ (or $\lesssim 1.0$) if
the magnitude limit of the survey can be $\sim 1-2$ (or $2-3$) magnitude
deeper. Applying this method to the SDSS DR7 QSOs with $z\lesssim 0.7$, we find
$\epsilon \propto M_{\bullet}^{0\sim 1.1}$ (or $\epsilon \propto
M_{\bullet}^{-1.0\sim 0}$) correlation for QSOs with the masses obtained
according to the H$\beta$ (or Mg II) empirical mass estimator. These
contradictory results may be due to the unknown systematic errors in the two
mass estimators, preventing an accurate constraint on the
$\epsilon-M_{\bullet}$ relation by using current available QSO samples. We find
that both the estimates of MBH mass and Eddington ratio distribution functions
can be affected by the $\epsilon-M_\bullet$ relation, suggesting that the
determination of this relation is important for understanding the accretion and
growth history of MBHs. In future, the intrinsic $\epsilon-M_{\bullet}$
relation is expected to be strongly constrained by using QSO samples obtained
from surveys deeper than SDSS if the host galaxy contamination and systematic
errors of the mass estimator(s) can be well modeled or removed.
