Model averaging for multiple quantile regression with covariates missing at random

In this paper, we develop a model averaging estimation procedure for multiple quantile regression where missingness occurs to the covariates. Our concern is on the improvement of prediction accuracy for multiple quantiles of response conditional on observed covariates. A set of candidate models is constructed according to missingness data patterns. In this model set, one model is based on the subjects with complete-case data, and the remaining models are based on the subsets of covariates with observed data. The weights for our model averaging are determined by a leave-one-out cross-validation criterion that is minimized over the complete case datasets. Under certain regularity conditions, we establish the asymptotic optimality for the selected weights in the sense of minimizing the out-of-sample combined quantile prediction error. Simulation studies are presented to demonstrate the advantages of the proposed approach vs. several existing active methods. As an illustration, a dataset from NHANES 2005-2006 is analysed.