Two-Stage Online Debiased Lasso Estimation and Inference for High-Dimensional Quantile Regression with Streaming Data

In this paper, the authors propose a two-stage online debiased lasso estimation and statistical inference method for high-dimensional quantile regression (QR) models in the presence of streaming data. In the first stage, the authors modify the QR score function based on kernel smoothing and obtain the online lasso smoothed QR estimator through iterative algorithms. The estimation process only involves the current data batch and specific historical summary statistics, which perfectly accommodates to the special structure of streaming data. In the second stage, an online debiasing procedure is carried out to eliminate biases caused by the lasso penalty as well as the accumulative approximation error so that the asymptotic normality of the resulting estimator can be established. The authors conduct extensive numerical experiments to evaluate the performance of the proposed method. These experiments demonstrate the effectiveness of the proposed method and support the theoretical results. An application to the Beijing PM2.5 Dataset is also presented.