Nonparametric Regression With Predictors Missing at Random

Abstract

Nonparametric regression with predictors missing at random (MAR), where the probability of missing depends only on observed variables, is considered. Univariate predictor is the primary case of interest. A new adaptive orthogonal series estimator is developed. Large sample theory shows that the estimator is rate-minimax and it is also sharp-minimax whenever predictors are missing completely at random (MCAR). Furthermore, confidence bands, estimation of nuisance functions, including conditional probability of observing the predictor, design density and scale, and multiple regression are also considered. Numerical study and a real example show feasibility of the proposed methodology for small samples. Supplementary materials, containing results of the numerical study, are available online.