# When is it worthwhile to jackknife? Breaking the quadratic barrier for Z-estimators

arXiv: https://arxiv.org/abs/2411.02909 · HTML: https://mlsanity.com/paper/2411.02909/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Nov 2024.

## Built on by (most cited first, 2 of 2)

- [Error Bounds for Statistical Estimators in BTL Model with Parametric Multivariate Utility Functions](https://mlsanity.com/paper/2609.26326/) (2026)
- [Stabilized Higher-Order Influence Functions: Statistical Theory of a Class of Bilinear Forms](https://mlsanity.com/paper/2607.04743/) (2026)

## Abstract (arXiv)

Resampling methods are especially well-suited to inference with estimators that provide only "black-box'' access. Jackknife is a form of resampling, widely used for bias correction and variance estimation, that is well-understood under classical scaling where the sample size n grows for a fixed problem. We study its behavior in application to estimating functionals using high-dimensional Z-estimators, allowing both the sample size n and problem dimension d to diverge. We begin showing that the plug-in estimator based on the Z-estimate suffers from a quadratic breakdown: while it is √n-consistent and asymptotically normal whenever n gtrsim d2, it fails for a broad class of problems whenever n lesssim d2. We then show that under suitable regularity conditions, applying a jackknife correction yields an estimate that is √n-consistent and asymptotically normal whenever ngtrsim d3/2. This provides strong motivation for the use of jackknife in high-dimensional problems where the dimension is moderate relative to sample size. We illustrate consequences of our general theory for various specific Z-estimators, including non-linear functionals in linear models; generalized linear models; and the inverse propensity score weighting (IPW) estimate for the average treatment effect, among others.