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 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 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) impose no assumptions on the marginal density , and are not obtained as a ratio between two kernel smoothing estimators which may sound like a go to approach in this problem.
Funding Statement
SB was partially supported by NSF grants DMS-1713003 and CCF-1763734. MN and SB were partially supported by NSF DMS-2113684.
Citation
Michael Li. Matey Neykov. Sivaraman Balakrishnan. "Minimax optimal conditional density estimation under total variation smoothness." Electron. J. Statist. 16 (2) 3937 - 3972, 2022. https://doi.org/10.1214/22-EJS2037
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