Two-step online estimation and inference for expected shortfall regression with streaming data

Expected shortfall, is defined as the average over the tail below (or above) a certain quantile of a probability distribution. The expected shortfall regression (ESR) provides powerful tools for analysing how the covariates affects the tail behaviour of a response variable, thus has drawn widespread attention. Due to the huge storage burden and the privacy limit to re-accessing the historical dataset, traditional methods for ESR are no longer applicable in many application scenarios. In this article, we develop an online two-step ESR estimator based on a Neyman-orthogonal score and the kernel smooth technique, which can be updated only using the current data batch and some summary statistics of historical data. Under some regularity conditions, we establish the non-asymptotic error bounds for the renewable estimators at each step. Moreover, it shows that the online ESR estimator attains the same asymptotic distribution as the offline ESR estimator. Numerical studies and a real data example are provided to evaluate the performance of our proposed method.