Bagging is a useful method for large-scale statistical analysis, especially when the computing resources are very limited. We study here the asymptotic properties of bagging estimators for $M$-estimation problems but with massive datasets. We theoretically prove that the resulting estimator is consistent and asymptotically normal under appropriate conditions. The results show that the bagging estimator can achieve the optimal statistical efficiency, provided that the bagging subsample size and the number of subsamples are sufficiently large. Moreover, we derive a variance estimator for valid asymptotic inference. All theoretical findings are further verified by extensive simulation studies. Finally, we apply the bagging method to the US Airline Dataset to demonstrate its practical usefulness.

翻译：Bagging是一种在大规模统计分析中非常有效的方法，尤其当计算资源极为有限时。本文研究了大容量数据集下$M$估计问题中Bagging估计量的渐近性质。我们从理论上证明，在适当条件下，该估计量具有相合性和渐近正态性。结果表明，只要Bagging子样本容量和子样本数量足够大，该估计量即可达到最优的统计效率。此外，我们推导了用于有效渐近推断的方差估计量。所有理论发现均通过广泛的模拟研究得到进一步验证。最后，我们将Bagging方法应用于美国航空公司数据集，以展示其实际应用价值。