Semi-supervised inference for the high-dimensional quantile regression

In this paper, we study a high-dimensional quantile working model under a semi-supervised setting, where a relatively small-size labeled data and a large-size of unlabeled data are available.We propose a semi-supervised debiased least absolute shrinkage and selection operator (LASSO) method to estimate the target parameter in a model-free framework. Under suitable conditions, we establish the limiting properties of the proposed semi-supervised estimator. Furthermore, we demonstrate that the proposed semi-supervised estimator is more efficient than the supervised estimator under model misspecification and achieves equivalent efficiency when the working model is correctly specified. Simulation studies are conducted to examine the finite-sample performance of the proposed method. An application is illustrated with an analysis of an electronic health record dataset.