  We apply a simple statistical method (Derenzo & Hildebrand 1969) to
estimating the completeness of quasar surveys. It requires that an area has
been covered by two or more, preferably different, selection techniques. We use
three suitable data sets with separate selections from: variability and
UV-excess (170 quasars); objective prism and UV-excess (141 quasars);
multicolour and X-ray ({\it ROSAT,} 19 quasars). We find that, for selection by
UV-excess, the common limit of $U-B \le -0.35 \pm -0.05$ leads to losses of
$\sim 35%$, typically missing low-luminosity $(M_{B} \gtrsim -24.5)$ quasars,
independently of redshift. Systematic incompleteness will therefore affect the
new generation of large quasar surveys that select by $U-B \le -0.35$. By
correcting for this incompleteness, we find, from the first data set ($B <
21.0$ and $z < 2.2$), that the evolution of the quasar luminosity function (LF)
is best described by joint luminosity and density evolution. When extrapolated
to $z = 0$, the LF matches that of local Seyfert galaxies better than any
previous determination. The LF shows an increase in the number of
low-luminosity quasars at low redshifts and of brighter quasars at intermediate
redshifts, relative to the LF of Boyle et al. (1990). This result is consistent
with models in which quasars fade from an initial bright phase.
