Minimax optimal conditional density estimation under total variation smoothness

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