  We present a new algorithm to estimate quasar photometric redshifts
(photo-$z$s), by considering the asymmetries in the relative flux distributions
of quasars. The relative flux models are built with multivariate Skew-t
distributions in the multi-dimensional space of relative fluxes as a function
of redshift and magnitude. For 151,392 quasars in the SDSS, we achieve a
photo-$z$ accuracy, defined as the fraction of quasars with the difference
between the photo-$z$ $z_p$ and the spectroscopic redshift $z_s$, $|\Delta z| =
|z_s-z_p|/(1+z_s)$ within 0.1, of 74%. Combining the WISE W1 and W2 infrared
data with the SDSS data, the photo-$z$ accuracy is enhanced to 87%. Using the
Pan-STARRS1 or DECaLS photometry with WISE W1 and W2 data, the photo-$z$
accuracies are 79% and 72%, respectively. The prior probabilities as a function
of magnitude for quasars, stars and galaxies are calculated respectively based
on (1) the quasar luminosity function; (2) the Milky Way synthetic simulation
with the Besan\c{c}on model; (3) the Bayesian Galaxy Photometric Redshift
estimation. The relative fluxes of stars are obtained with the Padova
isochrones, and the relative fluxes of galaxies are modeled through galaxy
templates. We test our classification method to select quasars using the DECaLS
$g$, $r$, $z$, and WISE W1 and W2 photometry. The quasar selection completeness
is higher than 70% for a wide redshift range $0.5<z<4.5$, and a wide magnitude
range $18<r<21.5$ mag. Our photo-$z$ regression and classification method has
the potential to extend to future surveys. The photo-$z$ code will be publicly
available.
