Jump-Preserving Estimation and Jump Detection for Nonparametric Model with Missing Covariate

Nonparametric regression analysis has broad applications. In some cases, the regression function with jumps (i.e., the regression curve is discontinuous) seems to be more appropriate to describe the related phenomena. Existing a number of methods for estimating discontinuous curve, most of which are based on the data is complete, which is unrealistic in many practical situations. In this paper, we consider estimating discontinuous nonparametric model with missing covariate data. Based on inverse selection probability weighted and jump-preserving techniques, a jump-preserving estimation procedure is proposed. The proposed method is capable of automatically accommodating possible jumps in the nonparametric function, without the requirement of prior knowledge regarding the number and locations of jump points. The proposed estimator for the discontinuous regression function is shown to be oracally efficient in the sense that it is uniformly indistinguishable from that when the selection probabilities are known. Furthermore, it is proved that the fitted curve by this procedure is consistent in the entire design space. Numerical simulation also indicates the performance of finite sample of this method is efficient and reliable.