放回抽样与Poisson抽样最优子抽样的比较研究
Sampling with replacement vs Poisson sampling: a comparative study in optimal subsampling
摘要：
面对海量数据，子采样是提高计算效率的常用技术，而采用非均匀子采样概率是提高估计效率的有效途径。为了计算效率，通常使用替换或通过泊松子采样来实现子采样。然而，没有严格的调查已经进行了研究之间的差异，这两个二次抽样程序，如他们的估计效率和计算方便。本文对这两种不同的抽样方法进行了比较研究。在最大化一般目标函数的背景下，我们首先推导出从两个抽样程序获得的估计量的渐近分布。结果表明，泊松子采样可能具有更高的估计效率。基于替换子采样和泊松子采样的渐近分布，我们得到最优子采样概率，最小化子采样估计量的方差函数。这些子采样概率进一步揭示了替换子采样和泊松子采样之间的异同。通过对两种二次抽样方法的理论描述和比较，为在实际应用中选择更合适的二次抽样方法提供了指导。在此基础上，提出了基于优化结果的实用算法，并通过理论和实证分析对算法进行了评价。
英文摘要：
Faced with massive data, subsampling is a commonly used technique to improve computational efficiency, and using nonuniform subsampling probabilities is an effective approach to improve estimation efficiency. For computational efficiency, subsampling is often implemented with replacement or through Poisson subsampling. However, no rigorous investigation has been performed to study the difference between the two subsampling procedures such as their estimation efficiency and computational convenience. This paper performs a comparative study on these two different sampling procedures. In the context of maximizing a general target function, we first derive asymptotic distributions for estimators obtained from the two sampling procedures. The results show that the Poisson subsampling may have a higher estimation efficiency. Based on the asymptotic distributions for both subsampling with replacement and Poisson subsampling, we derive optimal subsampling probabilities that minimize the variance functions of the subsampling estimators. These subsampling probabilities further reveal the similarities and differences between subsampling with replacement and Poisson subsampling. The theoretical characterizations and comparisons on the two subsampling procedures provide guidance to select a more appropriate subsampling approach in practice. Furthermore, practically implementable algorithms are proposed based on the optimal structural results, which are evaluated through both theoretical and empirical analyses.
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