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Faced with massive data, subsampling is a commonly used technique to improvecomputational efficiency, and using nonuniform subsampling probabilities is aneffective approach to improve estimation efficiency. For computationalefficiency, subsampling is often implemented with replacement or throughPoisson subsampling. However, no rigorous investigation has been performed tostudy the difference between the two subsampling procedures such as theirestimation efficiency and computational convenience. This paper performs acomparative study on these two different sampling procedures. In the context ofmaximizing a general target function, we first derive asymptotic distributionsfor estimators obtained from the two sampling procedures. The results show thatthe Poisson subsampling may have a higher estimation efficiency. Based on theasymptotic distributions for both subsampling with replacement and Poissonsubsampling, we derive optimal subsampling probabilities that minimize thevariance functions of the subsampling estimators. These subsamplingprobabilities further reveal the similarities and differences betweensubsampling with replacement and Poisson subsampling. The theoreticalcharacterizations and comparisons on the two subsampling procedures provideguidance to select a more appropriate subsampling approach in practice.Furthermore, practically implementable algorithms are proposed based on theoptimal structural results, which are evaluated through both theoretical andempirical analyses.

Sampling with replacement vs Poisson sampling: a comparative study in optimal subsampling