On nonparametric functional data regression with incomplete observations

In this work we consider the problem of nonparametric estimation of a regression function $m\left(\chi \right)=E\left(Y\right|\chi =\chi )$ with the functional covariate $\chi$ when the response $Y$ may be missing according to a missing-not-at-random (MNAR) setup, i.e., when the underlying missing probability mechanism can depend on both $\chi$ and $Y$. Our proposed estimator is based on a particular representation of the regression function $m\left(\chi \right)$ in terms of four associated conditional expectations that can be estimated nonparametrically. To assess the theoretical performance of our estimators, we study their convergence properties in general $L^p$ norms where we also look into their rates of convergence. Our numerical results show that the proposed estimators have good finite-sample performance. We also explore the applications of our results to the problem of statistical classification with missing labels and establish a number of convergence results for new kernel-type classification rules.