# Logistic Regression for Massive Data with Rare Events

arXiv: https://arxiv.org/abs/2006.00683 · HTML: https://mlsanity.com/paper/2006.00683/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2020.

## Built on by (most cited first, 9 of 9)

- [Optimal Binary Classification Beyond Accuracy](https://mlsanity.com/paper/2107.01777/) (2021)
- [Nonuniform Negative Sampling and Log Odds Correction with Rare Events Data](https://mlsanity.com/paper/2110.13048/) (2021)
- [Concentration and excess risk bounds for imbalanced classification with synthetic oversampling](https://mlsanity.com/paper/2510.20472/) (2025)
- [Optimal Downsampling for Imbalanced Classification with Generalized Linear Models](https://mlsanity.com/paper/2410.08994/) (2024)
- [Graph-Based Model-Agnostic Data Subsampling for Recommendation Systems](https://mlsanity.com/paper/2305.16391/) (2023)
- [A Provably Accurate Randomized Sampling Algorithm for Logistic Regression](https://mlsanity.com/paper/2402.16326/) (2024)
- [A Comparison of Approaches for Imbalanced Classification Problems in the Context of Retrieving Relevant Documents for an Analysis](https://mlsanity.com/paper/2205.01600/) (2022)
- [Scalable Counterfactual Risk Estimation for Rare Events in Longitudinal Data](https://mlsanity.com/paper/2606.01539/) (2026)
- [Scale-invariant Optimal Sampling for Rare-events Data with Sparse Models](https://mlsanity.com/paper/2608.22597/) (2026)

## Abstract (arXiv)

This paper studies binary logistic regression for rare events data, or imbalanced data, where the number of events (observations in one class, often called cases) is significantly smaller than the number of nonevents (observations in the other class, often called controls). We first derive the asymptotic distribution of the maximum likelihood estimator (MLE) of the unknown parameter, which shows that the asymptotic variance convergences to zero in a rate of the inverse of the number of the events instead of the inverse of the full data sample size. This indicates that the available information in rare events data is at the scale of the number of events instead of the full data sample size. Furthermore, we prove that under-sampling a small proportion of the nonevents, the resulting under-sampled estimator may have identical asymptotic distribution to the full data MLE. This demonstrates the advantage of under-sampling nonevents for rare events data, because this procedure may significantly reduce the computation and/or data collection costs. Another common practice in analyzing rare events data is to over-sample (replicate) the events, which has a higher computational cost. We show that this procedure may even result in efficiency loss in terms of parameter estimation.