On the Subbagging Estimation for Massive Data

TL;DR

This paper proposes a subbagging estimation method for large datasets that meets memory constraints, providing theoretical guarantees of consistency and normality, and demonstrating practical efficiency through simulations and real data analysis.

Contribution

It introduces a novel subbagging approach with a new theoretical framework for analyzing its properties under memory constraints, achieving $\, ext{sqrt}(N)$-consistency and asymptotic normality.

Findings

Subbagging estimator achieves $\, ext{sqrt}(N)$-consistency and asymptotic normality.

The asymptotic variance inflates by a factor of $1/\alpha$ compared to full sample estimator.

Simulation and real data analysis confirm the estimator's efficiency and accuracy.

Abstract

This article introduces subbagging (subsample aggregating) estimation approaches for big data analysis with memory constraints of computers. Specifically, for the whole dataset with size $N$, $m_N$ subsamples are randomly drawn, and each subsample with a subsample size $k_N\ll N$ to meet the memory constraint is sampled uniformly without replacement. Aggregating the estimators of $m_N$ subsamples can lead to subbagging estimation. To analyze the theoretical properties of the subbagging estimator, we adapt the incomplete $U$-statistics theory with an infinite order kernel to allow overlapping drawn subsamples in the sampling procedure. Utilizing this novel theoretical framework, we demonstrate that via a proper hyperparameter selection of $k_N$ and $m_N$, the subbagging estimator can achieve $\sqrt{N}$-consistency and asymptotic normality under the condition $(k_Nm_N)/N\to \alpha \in (0,\infty]$. Compared to the full sample estimator, we theoretically show that the $\sqrt{N}$-consistent subbagging estimator has an inflation rate of $1/\alpha$ in its asymptotic variance. Simulation experiments are presented to demonstrate the finite sample performances. An American airline dataset is analyzed to illustrate that the subbagging estimate is numerically close to the full sample estimate, and can be computationally fast under the memory constraint.