Asymptotic results for nonparametric regression estimators after sufficient dimension reduction estimation

Prediction, in regression and classification, is one of the main aims in modern data science. When the number of predictors is large, a common first step is to reduce the dimension of the data. Sufficient dimension reduction (SDR) is a well-established paradigm of reduction that keeps all the relevant information in the covariates X that is necessary for the prediction of Y. In practice, SDR has been successfully used as an exploratory tool for modeling after estimation of the sufficient reduction. Nevertheless, even if the estimated reduction is a consistent estimator of the population, there is no theory supporting this step when nonparametric regression is used in the imputed estimator. In this paper, we show that the asymptotic distribution of the nonparametric regression estimator remains unchanged whether the true SDR or its estimator is used. This result allows making inferences, for example, computing confidence intervals for the regression function, thereby avoiding the curse of dimensionality.