Renewable quantile regression for streaming data sets

Online updating is an important statistical method for the analysis of big data arriving in streams due to its ability to break the storage barrier and the computational barrier under certain circumstances. The quantile regression, as a widely used regression model in many fields, faces challenges in model fitting and variable selection with big data arriving in streams. Chen et al. (2019, Annals of Statistics) has proposed a quantile regression method for streaming data, but a strong additional condition is required. In this paper, renewable optimized objective functions for regression parameter estimation and variable selection in a quantile regression are proposed. The proposed methods are illustrated using current data and the summary statistics of historical data. Theoretically, the proposed statistics are shown to have the same asymptotic distributions as the standard version computed on an entire data stream with the data batches pooled into one data set, without additional condition. Both simulations and data analysis are conducted to illustrate the finite sample performance of the proposed methods.