# Predictive inference with the jackknife+

**Authors:** Rina Foygel Barber, Emmanuel J. Candes, Aaditya Ramdas, Ryan J. Tibshirani  
**arXiv:** [1905.02928](https://arxiv.org/abs/1905.02928)  
**Date:** 2019-05-08  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/1905.02928v1/

## Key points

- The paper introduces a method called the jackknife+ for constructing predictive confidence intervals.
- The jackknife+ incorporates leave-one-out predictions at the test point to account for variability in the fitted regression function.
- Unlike the original jackknife, the jackknife+ provides more accurate and reliable coverage guarantees by considering quantiles of leave-one-out residuals to determine interval width.
- The modified jackknife+ method ensures rigorous coverage guarantees regardless of the distribution of data points, which is significant for algorithms that treat training points symmetrically.
- Examples are provided where the coverage rate may vanish with the original jackknife approach.
- Theoretical and empirical analyses show that both the jackknife and jackknife+ intervals achieve nearly exact coverage and have similar lengths under stable fitting algorithms.
- The application of the jackknife+ is extended to K-fold cross-validation, with rigorous coverage properties established in this context as well.
- The proposed methods are related to cross-conformal prediction techniques introduced by Vovk (2015), and connections between these approaches are discussed.

## Summary

This paper introduces a novel method called the jackknife+ for constructing predictive confidence intervals. The jackknife+ builds upon the original jackknife method by incorporating leave-one-out predictions at the test point to account for variability in the fitted regression function. Unlike the original jackknife, which only considers quantiles of leave-one-out residuals to determine interval width, the jackknife+ provides more accurate and reliable coverage guarantees. The authors demonstrate that assuming exchangeable training samples, the modified jackknife+ method ensures rigorous coverage guarantees regardless of the distribution of data points. This is particularly significant for algorithms that treat training points symmetrically. In contrast, such guarantees are not possible with the original jackknife approach, and examples are provided where the coverage rate may actually vanish. Theoretical and empirical analyses conducted by the authors reveal that both the jackknife and jackknife+ intervals achieve nearly exact coverage and have similar lengths when the fitting algorithm exhibits stability. Additionally, they extend the application of the jackknife+ to K-fold cross-validation and establish rigorous coverage properties in this context as well. The proposed methods presented in this paper are related to cross-conformal prediction techniques introduced by Vovk (2015), and connections between these approaches are discussed. Overall, this paper contributes a valuable enhancement to predictive inference through its introduction of the jackknife+ method. By accounting for variability in fitted regression functions using leave-one-out predictions, it provides improved coverage guarantees compared to traditional methods like the original jackknife. The theoretical analysis and empirical evidence presented support its effectiveness in achieving nearly exact coverage with similar interval lengths under stable fitting algorithms.