  This paper studies the minimax rate of nonparametric conditional density
estimation under a weighted absolute value loss function in a multivariate
setting. We first demonstrate that conditional density estimation is impossible
if one only requires that $p_{X|Z}$ is smooth in $x$ for all values of $z$.
This motivates us to consider a sub-class of absolutely continuous
distributions, restricting the conditional density $p_{X|Z}(x|z)$ to not only
be H\"older smooth in $x$, but also be total variation smooth in $z$. We
propose a corresponding kernel-based estimator and prove that it achieves the
minimax rate. We give some simple examples of densities satisfying our
assumptions which imply that our results are not vacuous. Finally, we propose
an estimator which achieves the minimax optimal rate adaptively, i.e., without
the need to know the smoothness parameter values in advance. Crucially, both of
our estimators (the adaptive and non-adaptive ones) impose no assumptions on
the marginal density $p_Z$, and are not obtained as a ratio between two kernel
smoothing estimators which may sound like a go to approach in this problem.
