Optimal subsampling estimation of multiplicative models based on the least squared relative error criterion

Multiplicative regression is a useful alternative tool for modeling positive response variable. This paper investigates the optimal subsampling for multiplicative models based on the least squared relative error criterion. Initially, the consistency and asymptotic normality of the estimator from the general subsampling algorithm are demonstrated. Subsequently, the optimal subsampling probability is determined according to the L-optimality criterion. Based on the derived probability, a two-step algorithm is proposed: first, the uniform subsampling probability is used in a general subsampling algorithm to obtain an initial estimation; then, the optimal subsampling probability is calculated based on this initial estimation, and subsequently used for sampling to obtain the final estimation. We also establish the consistency and asymptotic normality of the proposed estimators. Numerical studies and two real data analyses confirm the efficiency of the proposed method.