Abstract: Subsampling techniques can effectively reduce the computational costs of processing big data. Practical subsampling plans typically involve initial uniform sampling and refined sampling. Subsample-based big data inferences are generally built on the inverse probability weighting (IPW), which may be unstable and cannot incorporate auxiliary information. In this paper, we consider a two-step Poisson sampling, which combines an initial uniform sampling with a second Poisson sampling. Under this sampling plan, we propose an empirical likelihood weighting (ELW) estimation approach to an M-estimation parameter, and then construct a nearly optimal two-step Poisson sampling plan based on the ELW method to improve estimation efficiency of IPW-based optimal subsamplings. Further, we derive methods for determining the smallest sample sizes with which the proposed sampling-and-estimation method produces estimators of guaranteed precision. Our ELW method overcomes the instability of IPW by circumventing the use of inverse probabilities, and utilizes auxiliary information including the size and certain sample moments of big data. We show that the proposed ELW method produces more efficient estimators than IPW, leading to more efficient optimal sampling plans and more economical sample sizes for a prespecified estimation precision. These advantages are confirmed through real data based simulations.

Key words and phrases: Big data, empirical likelihood, two-step Poisson sampling.