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

Abstract

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the huge storage burden and the privacy limit to re-accessing the historical dataset, traditional methods for ESR are no longer applicable in many application scenarios.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.

Keywords:

• Expected shortfall regression
• neyman-orthogonal
• kernel smooth
• streaming data

Acknowledgments

The authors are grateful to the Editor, an Associate Editor and three anonymous referees for their insightful comments and suggestions on this article, which have led to significant improvements. All authors contributed to this work equally.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Our research was supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China [grant number 12271272].