  Redshift is a key quantity for inferring cosmological model parameters. In
photometric redshift estimation, cosmologists use the coarse data collected
from the vast majority of galaxies to predict the redshift of individual
galaxies. To properly quantify the uncertainty in the predictions, however, one
needs to go beyond standard regression and instead estimate the full
conditional density f(z|x) of a galaxy's redshift z given its photometric
covariates x. The problem is further complicated by selection bias: usually
only the rarest and brightest galaxies have known redshifts, and these galaxies
have characteristics and measured covariates that do not necessarily match
those of more numerous and dimmer galaxies of unknown redshift. Unfortunately,
there is not much research on how to best estimate complex multivariate
densities in such settings. Here we describe a general framework for properly
constructing and assessing nonparametric conditional density estimators under
selection bias, and for combining two or more estimators for optimal
performance. We propose new improved photo-z estimators and illus- trate our
methods on data from the Sloan Data Sky Survey and an application to
galaxy-galaxy lensing. Although our main application is photo-z estimation, our
methods are relevant to any high-dimensional regression setting with
complicated asymmetric and multimodal distributions in the response variable.
