  A scaling relation based on thin-disc accretion theory has been used by some
workers to determine the mass-inflow rate onto 20 high-redshift (z) and 80
Palomar-Green quasars. Based on several assumptions, it inexplicably implies
that the inflow rate is an inverse function of black-hole (BH) mass Mbh.
Moreover, its results remain untested. This paper offers a simple empirical
relation essentially free of assumptions, found using available data for 59
highest-z quasars and the so-called Salpeter relation. We find that the
accretion rate is proportional to Mbh(1+z)3, consistent with conventional
astrophysics that the accretion rate is a direct function of both Mbh and the
ambient gas density. We apply it to the 20 high-z and a subset of Palomar-Green
quasars. Comparative analyses show that all empirically derived accretion rates
and radiative efficiencies pass the tests, but their theoretical counterparts
fail in most cases. A secondary relation defines the Eddington ratio as a
function of z and radiative efficiency. Consistent with the empirical
relations, spline regression analysis of Kozlowski's data for 132,000 quasars
at z<2.4 shows that both the Eddington ratio and radiative efficiency are
functions of Mbh and z. The results show that bigger BHs accrete more
efficiently the smaller ones. For BHs > a billion solar masses, we get
radiative efficiency of ~0.23 at z>5.7 and ~0.84 at z<0.005. Notably, the
empirical relations predict a mass-inflow rate of 0.11-0,21 solar mass/year on
to the BH in M87 that matches its Bondi accretion rate determined using
observed density and temperature profiles.
