Optimal distributed subsampling for expected shortfall regression via Neyman-orthogonal score

Massive data bring a big challenge for analysis, and subsampling as an effective solution can significantly reduce the computational burden and maintain estimation efficiency. Expected Shortfall Regression (ESR) studies the impact of covariates on the tail expectation of response and explores the heterogeneous effects of the covariates. For joint linear quantile and expected shortfall regression models, we study the optimal subsampling method for ESR based on the Neyman-orthogonal score to reduce sensitivity with respect to nuisance parameters in quantile regression. When the massive data are stored in different sites, we further propose a distributed optimal subsampling method for the ESR. Asymptotic properties of the resultant estimators are established and the two-step algorithms are proposed for practical implementation. Extensive simulations and applications to Protein Tertiary Structure and Beijing Air Quality datasets show satisfactory performance of the proposed estimators. © 2025 Elsevier B.V.