  Density regression characterizes the conditional density of the response
variable given the covariates, and provides much more information than the
commonly used conditional mean or quantile regression. However, it is often
computationally prohibitive in applications with massive data sets, especially
when there are multiple covariates. In this paper, we develop a new data
reduction approach for the density regression problem using conditional support
points. After obtaining the representative data, we exploit the penalized
likelihood method as the downstream estimation strategy. Based on the
connections among the continuous ranked probability score, the energy distance,
the $L_2$ discrepancy and the symmetrized Kullback-Leibler distance, we
investigate the distributional convergence of the representative points and
establish the rate of convergence of the density regression estimator. The
usefulness of the methodology is illustrated by modeling the conditional
distribution of power output given multivariate environmental factors using a
large scale wind turbine data set. Supplementary materials for this article are
available online.
