  We investigate the Eddington ratio distribution of X-ray selected broad-line
active galactic nuclei (AGN) in the redshift range 1.0<z<2.2, where the number
density of AGNs peaks. Combining the optical and Subaru/FMOS near-infrared
spectroscopy, we estimate black hole masses for broad-line AGNs in the Chandra
Deep Field-South (CDF-S), Extended Chandra Deep Field-South (E-CDF-S), and the
XMM-Newton Lockman Hole (XMM-LH) surveys. AGNs with similar black hole masses
show a broad range of AGN bolometric luminosities, which are calculated from
X-ray luminosities, indicating that the accretion rate of black holes is widely
distributed. We find that a substantial fraction of massive black holes
accreting significantly below the Eddington limit at z~2, in contrast to what
is generally found for luminous AGNs at high redshift. Our analysis of
observational selection biases indicates that the "AGN cosmic downsizing"
phenomenon can be simply explained by the strong evolution of the co-moving
number density at the bright end of the AGN luminosity function, together with
the corresponding selection effects. However, it might need to consider a
correlation between the AGN luminosity and the accretion rate of black holes
that luminous AGNs have higher Eddington ratios than low-luminosity AGNs in
order to understand the relatively small fraction of low-luminosity AGNs with
high accretion rates in this epoch. Therefore, the observed downsizing trend
could be interpreted as massive black holes with low accretion rates, which are
relatively fainter than less massive black holes with efficient accretion.
