Moment-assisted GMM for Improving Subsampling-based MLE with Large-scale data

The maximum likelihood estimation is computationally demanding for large datasets, particularly when the likelihood function includes integrals. Subsampling can reduce the computational burden, but it typically results in efficiency loss. This paper proposes a moment-assisted subsampling (MAS) method that can improve the estimation efficiency of existing subsampling-based maximum likelihood estimators. The motivation behind this approach stems from the fact that sample moments can be efficiently computed even if the sample size of the whole data set is huge. Through the generalized method of moments, the proposed method incorporates informative sample moments of the whole data. The MAS estimator can be computed rapidly and is asymptotically normal with a smaller asymptotic variance than the corresponding estimator without incorporating sample moments of the whole data. The asymptotic variance of the MAS estimator depends on the specific sample moments incorporated. We derive the optimal moment that minimizes the resulting asymptotic variance in terms of Loewner order. Simulation studies and real data analysis were conducted to compare the proposed method with existing subsampling methods. Numerical results demonstrate the promising performance of the MAS method across various scenarios.