On the $$L_p$$ norms of kernel regression estimators for incomplete data with applications to classification

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We consider kernel methods to construct nonparametric estimators of a regression function based on incomplete data. To tackle the presence of incomplete covariates, we employ Horvitz–Thompson-type inverse weighting techniques, where the weights are the selection probabilities. The unknown selection probabilities are themselves estimated using (1) kernel regression, when the functional form of these probabilities are completely unknown, and (2) the least-squares method, when the selection probabilities belong to a known class of candidate functions. To assess the overall performance of the proposed estimators, we establish exponential upper bounds on the $$L_p$$ norms, $$1\le p<\infty $$ , of our estimators; these bounds immediately yield various strong convergence results. We also apply our results to deal with the important problem of statistical classification with partially observed covariates.