Doubly robust kernel density estimation when group membership is missing at random

When there are subjects with subpopulation memberships missing, the kernel density estimates of the subpopulations based on the subjects with verified memberships may not be valid unless the missingness of the memberships satisfies the stringent missing completely at random assumption. Under more plausible missing at random (MAR) assumptions, the inverse probability weighting (IPW) kernel smoothing method can be employed if the missing mechanism is well understood. On the other hand, if the memberships can be well predicted by a prediction model, one may replace the memberships (either all or only the missing ones), with the probabilities for the memberships to deal with the missing values. By integrating the information of the missing mechanism and the prediction model, we develop a doubly robust kernel density estimate which is valid as long as either of the models is correctly specified. Asymptotic properties of the new estimates are derived and the asymptotic orders of the mean integrated square errors (MISE) are compared with existing methods. We prove that the doubly robust estimate can be more efficient than the IPW estimate when the model for the missing mechanism is correctly specified. We also prove that the prediction-model based approaches may have smaller MISEs than the complete data when the prediction model is correctly specified, suggesting that those methods can be applied even when there are no missing values in the memberships. Simulation studies are carried out to assess the performances of methods and a real study example is used for illustrative purposes. (C) 2019 Elsevier B.V. All rights reserved.