Renewable estimation in linear expectile regression models with streaming data sets

Streaming data often exhibit heterogeneity due to heteroscedastic variances or inhomogeneous covariate effects. Online renewable quantile and expectile regression methods provide valuable tools for detecting such heteroscedasticity by combining current data with summary statistics from historical data. However, quantile regression can be computationally demanding because of the non-smooth check function. To address this, we propose a novel online renewable method based on expectile regression, which efficiently updates estimates using both current observations and historical summaries, thereby reducing storage requirements. By exploiting the smoothness of the expectile loss function, our approach achieves superior computational efficiency compared with existing online renewable methods for streaming data with heteroscedastic variances or inhomogeneous covariate effects. We establish the consistency and asymptotic normality of the proposed estimator under mild regularity conditions, demonstrating that it achieves the same statistical efficiency as oracle estimators based on full individual-level data. Numerical experiments and real-data applications demonstrate that our method performs comparably to the oracle estimator while maintaining high computational efficiency and minimal storage costs.