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The V-fold jackknife for semiparametric inference: variance estimation, confidence intervals, and simultaneous confidence bands

This paper introduces the V-fold jackknife as a computationally efficient and theoretically justified alternative to the bootstrap for semiparametric inference, establishing its validity for constructing confidence intervals and simultaneous bands across both standard and generalized asymptotically linear estimators without requiring influence function derivation.

Original authors: Yi Li, Ashkan Ertefaie, Mark van der Laan

Published 2026-07-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yi Li, Ashkan Ertefaie, Mark van der Laan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery: "How sure can we be about our answer?" In the world of statistics, this is called inference. When scientists use data to guess a truth—like the average effect of a new medicine or the survival rate of a patient—they need a way to measure how much their answer might wiggle if they collected different data. This wiggle room is called uncertainty, and the tool used to measure it is often a confidence interval. Think of a confidence interval as a safety net: if you say the answer is "50," a 95% confidence interval might say, "We are 95% sure the real answer is between 45 and 55."

For decades, the detective's favorite tool for building this safety net has been the bootstrap. Imagine you have a bag of marbles representing your data. The bootstrap says, "Let's reach in, grab a handful, write down the colors, put them back, and do it again." You repeat this thousands of times, creating thousands of fake datasets. By seeing how much the answers change across these thousands of tries, you can figure out how shaky your real answer is. It's powerful because it works for almost anything, but it's also exhausting. If your math problem is complex (like those used in modern machine learning), running it thousands of times can take forever, like trying to count every grain of sand on a beach by picking them up one by one.

Recently, a new problem has emerged. In the age of artificial intelligence and complex algorithms, the "bag of marbles" trick sometimes breaks. When you pull marbles out and put them back, you might accidentally pick the same marble twice, or miss some entirely. For simple math, this doesn't matter. But for the fancy, adaptive algorithms used today, this tiny glitch can mess up the whole safety net, making the confidence intervals lie about how safe they are. Scientists needed a new tool that was both fast enough to run on a laptop and smart enough to handle these tricky, modern algorithms without lying about the results.

This is where the paper "The V-Fold Jackknife for Semiparametric Inference" steps in. The authors, Yi Li, Ashkan Ertefaie, and Mark Van Der Laan, propose a clever alternative called the V-Fold Jackknife. Instead of the "grab-and-replace" game of the bootstrap, they suggest a "divide-and-conquer" strategy. Imagine you have a giant pizza (your data). Instead of making thousands of fake pizzas, you simply cut the real pizza into V slices (folds). You then take one slice away, solve the puzzle with the remaining slices, and write down the answer. You do this for every slice, so you end up with V different answers.

The magic of this method is in how it uses those answers. The authors show that by looking at how much these V answers differ from each other, you can build a safety net that is just as trustworthy as the one built by the thousands of bootstrap tries, but it only requires you to solve the puzzle V times (usually between 5 and 20 times). That's a massive speedup.

But here is the really cool part: the paper proves that this method works even when the math gets weird. Usually, when you have a small number of slices (a small V), you'd expect your safety net to be wobbly. However, the authors discovered that if you use a specific type of mathematical "ruler" (called a t-distribution with V-1 degrees of freedom) to measure the wiggle room, the safety net stays strong. It's like having a ruler that automatically gets longer and more cautious when you have fewer slices to measure, ensuring you don't accidentally step off a cliff.

The paper also tackles a scenario where the "wiggle" gets bigger as you get more data, which happens in some advanced machine learning models. The V-Fold Jackknife handles this naturally because it measures the wiggle directly from the data slices, rather than trying to calculate a complex formula that might break.

To test their idea, the authors ran simulations on three different types of problems:

Average Treatment Effect: Figuring out if a treatment works. They found that while the old "influence function" method (a standard formula-based approach) often gave safety nets that were too small (under-covering), the V-Fold Jackknife kept the safety net wide enough to be reliable, even with small sample sizes.
Survival Curves: Tracking how long patients survive. Here, the V-Fold Jackknife performed just as well as the best existing methods but was much faster to compute.
Dose-Response Curves: This is where the old methods really struggled. In these complex scenarios, the standard formulas often crashed or gave wrong answers (failing up to 6.2% of the time in their tests). The V-Fold Jackknife, however, never failed and provided the most reliable safety nets of all.

The authors also showed how to use this method to create a "safety blanket" that covers an entire curve at once, not just a single point. They found that even with a modest number of slices (like 20), the method works surprisingly well, especially because the data in these problems often has a hidden simplicity (low "effective rank") that the method can exploit.

In short, this paper introduces a tool that is like a high-speed, low-maintenance safety net. It doesn't require the heavy lifting of running thousands of simulations, and it doesn't break when the math gets complicated or the data behaves strangely. It offers a way for scientists to trust their answers in the age of machine learning, ensuring that when they say, "We are 95% sure," they really mean it.

Technical Summary: The V-Fold Jackknife for Semiparametric Inference

Problem Statement
The bootstrap has long served as the default tool for statistical inference due to its broad applicability and minimal analytic requirements. However, its theoretical validity for modern semiparametric and machine learning-based estimators remains largely unstudied. In contemporary settings involving high-dimensional nuisance estimation, adaptive model selection, and machine learning algorithms, the ordinary nonparametric bootstrap often fails. Resampling with replacement perturbs the empirical distribution in ways that alter the data-adaptive estimation path (e.g., tuning parameter selection, internal cross-validation splits, and fitted nuisance estimators). Consequently, the estimator computed on a bootstrap sample may not satisfy the same first-order expansion as the original estimator, leading to invalid inference. Furthermore, the computational cost of refitting complex estimators hundreds or thousands of times is substantial. There is a need for an inference framework that is computationally efficient, theoretically justified for asymptotically linear estimators, and does not rely on analytic derivation of influence functions.

Methodology
The paper proposes the V-fold jackknife (a grouped or delete-a-group jackknife) as a computationally efficient and theoretically justified alternative. The procedure involves:

Partitioning: Dividing the sample of size $n$ into $V$ disjoint, approximately equal-sized folds $\{I_1, \dots, I_V\}$ .
Leave-Fold-Out Estimation: Computing the estimator $\hat{\Psi}(P_{n,-v})$ on the subsample excluding fold $v$ , for each $v=1, \dots, V$ .
Pseudo-value Construction: Calculating jackknife pseudo-values defined as $IC_{Jack}(v) = V\hat{\Psi}(P_n) - (V-1)\hat{\Psi}(P_{n,-v})$ .
Variance Estimation: Estimating the variance using the empirical dispersion of these pseudo-values: $\hat{S}^2_{Jack} = (V-1)^{-1} \sum_{v=1}^V (IC_{Jack}(v) - \hat{\Psi}_{Jack}(P_n))^2$ .
Inference: Constructing confidence intervals using the Studentized statistic and critical values from the $t$ -distribution with $V-1$ degrees of freedom ( $t_{V-1}$ ), rather than the standard normal distribution.

The method is developed for regular asymptotically linear estimators of pathwise differentiable parameters (e.g., TMLE, one-step estimators, augmented IPW) and extended to generalized asymptotically linear estimators where the influence function variance diverges with sample size (e.g., HAL dose-response curves).

Key Contributions
The paper establishes six primary theoretical and methodological contributions:

Fixed-V Asymptotic Theory: The authors extend Brillinger's (1964) result from maximum likelihood estimators to the full class of regular asymptotically linear estimators. They prove that for a fixed $V \ge 2$ , the Studentized V-fold jackknife statistic converges in distribution to a $t_{V-1}$ distribution. This allows for valid confidence intervals even though the jackknife variance estimator does not converge in probability to the true asymptotic variance when $V$ is fixed.
Diverging-V Consistency: The paper establishes that as $V \to \infty$ (subject to $V = o(n)$ and specific remainder stability conditions), the jackknife variance estimator is consistent at rate $V^{-1/2}$ . This supports the use of Wald-type inference with $V$ growing slowly (e.g., $V = \log n$ ).
Simultaneous Confidence Bands: The authors develop the first simultaneous confidence band construction based on the grouped jackknife. This method utilizes a componentwise-Studentized limiting distribution that accounts for the Wishart structure of the fold-level denominators. Critical values are computed via Monte Carlo simulation from the estimated correlation matrix, avoiding the need for analytic influence function stacking or massive bootstrap refits.
Scale Invariance for Non- $\sqrt{n}$ Rates: The framework is extended to estimators with slower-than- $\sqrt{n}$ convergence rates where the influence function variance diverges (e.g., HAL estimators). The Studentization process is shown to be scale-invariant; the unknown diverging rate cancels out in the numerator and denominator, allowing valid inference without explicitly deriving or estimating the effective dimension or convergence rate.
Bias-Corrected Bands: The paper introduces bias-corrected simultaneous confidence bands that widen the interval by the absolute jackknife bias estimate ( $|\hat{b}_j|$ ) to address finite-sample bias, particularly relevant for non-smooth parameters.
Empirical Validation: Extensive simulations on average treatment effects (TMLE), Kaplan–Meier survival curves, and HAL-based dose-response curves demonstrate the method's reliability.

Results
Simulation studies confirm the theoretical properties of the V-fold jackknife:

Scalar Inference (ATE): In scenarios with severe positivity violations or misspecified nuisance models, standard influence-function-based (EIC) methods and robust variance estimators suffer from substantial undercoverage (as low as 70%). The V-fold jackknife with $t_{V-1}$ critical values maintains coverage near or above the nominal 95% level across all scenarios, even at small sample sizes ( $n=200$ ). While the intervals are wider than bootstrap intervals at small $V$ and $n$ , they provide reliable coverage where other methods fail.
Functional Inference (Survival Curves): For the Kaplan–Meier estimator, the V-fold jackknife produces pointwise and simultaneous bands comparable to established methods (Greenwood formula, Hall–Wellner, bootstrap). The method handles the singularity of the estimated correlation matrix (when the number of evaluation points $m$ exceeds $V-1$ ) effectively, leveraging the low effective rank of the correlation structure.
Non-Standard Rates (HAL Dose-Response): For HAL-based dose-response curves, the delta method frequently fails numerically (producing NA standard errors) and statistically (severe undercoverage, e.g., 19.2% for simultaneous bands in oscillating scenarios). The V-fold jackknife never fails numerically and consistently achieves the highest coverage among compared methods. The bias-corrected bands successfully restore near-nominal coverage in settings with moderate bias.

Significance
The paper claims that the V-fold jackknife offers a practical, computationally efficient, and theoretically grounded alternative to the bootstrap for modern semiparametric inference. Its significance lies in three areas:

Computational Efficiency: It requires only $V$ refits (e.g., 5–20) compared to the hundreds or thousands required by the bootstrap, making it feasible for expensive machine learning-based estimators.
Theoretical Robustness: It provides valid inference for asymptotically linear estimators without requiring the complex analytic derivation of influence functions or the verification of bootstrap validity conditions that are often violated by data-adaptive algorithms.
Versatility: It naturally handles non-standard convergence rates and diverging variances through scale invariance, a property that allows practitioners to perform inference on complex functionals (like dose-response curves) without knowing the effective dimension or convergence rate.

The authors emphasize that while the fixed- $V$ $t_{V-1}$ correction provides a finite-sample improvement over normal approximations, the method's validity for simultaneous bands and diverging- $V$ scenarios relies on the fold-level remainder stability of the underlying estimator. The approach is particularly valuable in settings where analytic standard errors are unavailable, difficult to derive, or numerically unstable.

Technical Summary: The V-Fold Jackknife for Semiparametric Inference

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