On the correct regression function (in L2) and its applications when the dimension of the covariate vector is random

We derive the optimal regression function (i.e., the best approximation in the L2 sense) when the vector of covariates has a random dimension. Furthermore, we consider applications of these results to problems in statistical regression and classification with missing covariates. It will be seen, perhaps surprisingly, that the correct regression function for the case with missing covariates can sometimes perform better than the usual regression function corresponding to the case with no missing covariates. This is because even if some of the covariates are missing, an indicator random variable δ, which is always observable, and is equal to 1 if there are no missing values (and 0 otherwise), may have far more information and predictive power about the response variable Y than the missing covariates do. We also propose kernel-based procedures for estimating the correct regression function nonparametrically. As an alternative estimation procedure, we also consider the least-squares method.