Nonparametric regression with responses missing at random

Abstract

For the case of a complete sample of univariate predictors and responses, the modern nonparametric regression matches results known for parametric and semiparametric regressions. The situation changes dramatically if some values in a sample are missing. This paper develops the theory of nonparametric regression for the classical case of responses missing at random. The main conclusion is that an adaptive estimator, based on a complete-case subsample, is asymptotically sharp minimax over all possible oracle-estimators that know: an underlying sample with missing responses; probability of observing the response given the predictor; smoothness of an underlying regression function; design density of the predictor; scale function of the regression error.