Do quasar broad-line velocity widths add any information to virial black
  hole mass estimates?
  We examine how much information measured broad-line widths add to virial BH
mass estimates for flux limited samples of quasars. We do this by comparing the
BH mass estimates to those derived by randomly reassigning the quasar
broad-line widths to different objects and re-calculating the BH mass. For 9000
BH masses derived from the H_beta line we find that the distributions of
original and randomized BH masses in the M_BH-redshift plane and the
M_BH-luminosity plane are formally identical. A 2D KS test does not find a
difference at >90% confidence. For the MgII line (32000 quasars) we do find
very significant differences between the randomized and original BH masses, but
the amplitude of the difference is still small. The difference for the CIV line
(14000 quasars) is 2-3sigma and again the amplitude of the difference is small.
Subdividing the data into redshift and luminosity bins we find that the median
absolute difference in BH mass between the original and randomized data is
0.025, 0.01 and 0.04 dex for H_beta, MgII and CIV respectively. The maximum
absolute difference is always <=0.1 dex. We investigate whether our results are
sensitive to corrections to MgII virial masses (e.g. Onken & Kollmeier 2008).
These corrections do not influence our results. Moreover, the mass residual -
Eddington ratio correlation discussed by Onken & Kollmeier is more directly
attributable to the slope of the relation between H_beta and MgII line width.
Our results imply that the measured quasar broad-line velocity widths provide
little extra information, after allowing for the mean velocity width. In this
case virial estimates are equivalent to M_BH L^alpha, with L/L_edd L^(1-alpha)
(with alpha~0.5). This leaves an unanswered question of why the accretion
efficiency changes with luminosity in just the right way to keep the mean
broad-line widths fixed as a function of luminosity. (abridged)
