A two-stage optimal subsampling estimation for missing data problems with large-scale data

Abstract

Subsampling is useful to downsize data volumes and speed up calculations for large-scale data and is well studied with completely observed data. In the presence of missing data, computation is more challenging and subsampling becomes more crucial and complex. However, there is still a lack of study on subsampling for missing data problems. This paper fills the gap by studying the subsampling method for a widely used missing data estimator, the augmented inverse probability weighting (AIPW) estimator. The response mean estimation problem with missing responses is discussed for illustration. A two-stage subsampling method is proposed via Poisson sampling framework. A small subsample of expected size n1 is used in the first stage to estimate the parameters in the propensity score and the outcome regression models, while a larger subsample of expected size n2 is used in the computationally simple second stage to calculate the final estimator. An attractive property of the resulting estimator is that its convergence rate is n2−1/2 rather than n1−1/2 when both the propensity score and the outcome regression functions are correctly specified. The rate n2−1/2 is still attainable for some important cases if only one of the two functions is correctly specified. This indicates that using a small subsample in the computationally complex first stage can reduce the computational burden with little impact on the statistical accuracy. Asymptotic normality of the resulting estimator is established and the optimal subsampling probability is derived by minimizing the asymptotic variance of the resulting estimator. Simulations and a real data analysis were conducted to demonstrate the empirical performance of the resulting estimator.