Multivariate adaptive warped kernel estimation

Abstract

We deal with the problem of nonparametric estimation of a multivariate regression function without any assumption on the compacity of the support of the random design. To tackle the problem, we propose to extend a “warping” device to the multivariate framework. An adaptive warped kernel estimator is first defined in the case of known design distribution and proved to be optimal in the oracle sense. Then, a general procedure is carried out: the marginal distributions of the design are estimated by the empirical cumulative distribution functions, and the dependence structure is built using a kernel estimation of the copula density. The copula density estimator is also studied and proved to be optimal in the oracle and in the minimax sense. The plug-in of this estimator in the regression function estimator provides a fully data-driven procedure. A numerical study illustrates the theoretical results.