Optimal Subsampling for Upper Expectation Parametric Regression

In classic regression analysis, the error term of the model typically conforms to the requirement of being independent and identically distributed. However, in the realm of big data, it is exceedingly common for the error term to exhibit varying distributions due to discrepancies in data collection timing and sources. In this article, we expand upon and refine the upper expectation parameter regression model, introducing a novel upper expectation loss form, which handles distribution heterogeneity via group-specific μi, to ensure consistent and efficient parameter estimation. Furthermore, we establish the asymptotic properties of this estimation. To address the challenges posed by big data or the restrictions of privacy, we propose a method utilizing Poisson subsampling to devise a new loss function. Under certain assumptions, this method satisfies the condition of asymptotic normality. Additionally, based on the asymptotic properties, we determine the optimal sampling probability and introduce the optimal subsampling technique. Our sampling method surpasses uniform sampling, which reduces MSE by about 50% compared to uniform sampling, and is straightforward to implement in practical scenarios. Subsequent simulation experiments and real-world examples further demonstrate the effectiveness of our approach.