This is the fourth paper in a series aimed at finding high-redshift quasars from five-color ( u ^ { \prime } g ^ { \prime } r ^ { \prime } i ^ { \prime } z ^ { \prime } ) imaging data taken along the Celestial Equator by the Sloan Digital Sky Survey ( SDSS ) during its commissioning phase .
In this paper , we use the color-selected sample of 39 luminous high-redshift quasars presented in Paper III to derive the evolution of the quasar luminosity function over the range of 3.6 < z < 5.0 , and -27.5 < M _ { 1450 } < -25.5 ( \Omega = 1 , H _ { 0 } = 50 km s ^ { -1 } Mpc ^ { -1 } ) .

We use the selection function derived in Paper III to correct for sample incompleteness .
The luminosity function is estimated using three different methods : ( 1 ) the 1 / V _ { a } estimator ; ( 2 ) a maximum likelihood solution , assuming that the density of quasars depends exponentially on redshift and as a power law in luminosity , and ( 3 ) Lynden-Bell ’ s non-parametric C ^ { - } estimator .
All three methods give consistent results .
The luminous quasar density decreases by a factor of \sim 6 from z = 3.5 to z = 5.0 , consistent with the decline seen from several previous optical surveys at z < 4.5 .
The luminosity function follows \psi ( L ) \propto L ^ { -2.5 } for z \sim 4 at the bright end , significantly flatter than the bright end luminosity function \psi ( L ) \propto L ^ { -3.5 } found in previous studies for z < 3 , suggesting that the shape of the quasar luminosity function evolves with redshift as well , and that the quasar evolution from z = 2 to 5 can not be described as pure luminosity evolution .
Possible selection biases and the effect of dust extinction on the redshift evolution of the quasar density are also discussed .