Minimax optimal conditional density estimation under total variation smoothness∗

Abstract

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 pX|Z is smooth in x for all values of z. This motivates us to consider a sub-class of absolutely con-tinuous distributions, restricting the conditional density pX|Z (x|z) to not only be Hölder 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) im-pose no assumptions on the marginal density pZ, and are not obtained as a ratio between two kernel smoothing estimators which may sound like a go to approach in this problem.