Optimal Poisson Subsampling for Multiplicative Regressions With Massive Data

Multiplicative regression is a pivotal tool for analyzing data with positive responses, such as stock prices and lifetimes, yet in the era of big data its parameter estimation confronts substantial computational challenges. The subsampling method offers an economical way to address the computational burden by extracting an informative small‐size subsample. However, optimal subsampling with replacement often results in duplicate observations, compromising estimation efficiency. In this paper, we propose an efficient Poisson subsampling method for large‐scale multiplicative regressions. Two oracle‐optimal subsampling probabilities based on A‐ and L‐optimality criteria, along with their practical implementation and a distributed computing‐adapted algorithm, are developed. Theoretically, we derive the properties of consistency, asymptotic normality, and concentration inequalities for the subsampling parameter estimator. The effectiveness of the proposed Poisson subsampling is demonstrated on synthetic and real datasets.