# The Influence Function of Semiparametric Estimators

arXiv: https://arxiv.org/abs/1508.01378 · HTML: https://mlsanity.com/paper/1508.01378/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Aug 2015.

## Built on by (most cited first, 30 of 36)

- [Robust and Agnostic Learning of Conditional Distributional Treatment Effects](https://mlsanity.com/paper/2205.11486/) (2022)
- [Inference on Strongly Identified Functionals of Weakly Identified Functions](https://mlsanity.com/paper/2208.08291/) (2022)
- [Simplifying debiased inference via automatic differentiation and probabilistic programming](https://mlsanity.com/paper/2405.08675/) (2024)
- [A Simple and General Debiased Machine Learning Theorem with Finite Sample Guarantees](https://mlsanity.com/paper/2105.15197/) (2021)
- [Automatic Debiased Machine Learning for Smooth Functionals of Nonparametric M-Estimands](https://mlsanity.com/paper/2501.11868/) (2025)
- [Adaptive Instrument Design for Indirect Experiments](https://mlsanity.com/paper/2312.02438/) (2023)
- [On Efficient Estimation of Distributional Treatment Effects under Covariate-Adaptive Randomization](https://mlsanity.com/paper/2506.05945/) (2025)
- [Learning Decision Policies with Instrumental Variables through Double Machine Learning](https://mlsanity.com/paper/2405.08498/) (2024)
- [Estimating Distributional Treatment Effects in Randomized Experiments: Machine Learning for Variance Reduction](https://mlsanity.com/paper/2407.16037/) (2024)
- [Multiply Robust Causal Mediation Analysis with Continuous Treatments](https://mlsanity.com/paper/2105.09254/) (2021)
- [Optimal Best Arm Identification in Two-Armed Bandits with a Fixed Budget under a Small Gap](https://mlsanity.com/paper/2201.04469/) (2022)
- [A Free Lunch with Influence Functions? Improving Neural Network Estimates with Concepts from Semiparametric Statistics](https://mlsanity.com/paper/2202.09096/) (2022)
- [Doubly Robust Proximal Causal Learning for Continuous Treatments](https://mlsanity.com/paper/2309.12819/) (2023)
- [Automated Efficient Estimation using Monte Carlo Efficient Influence Functions](https://mlsanity.com/paper/2403.00158/) (2024)
- [Best Arm Identification with Contextual Information under a Small Gap](https://mlsanity.com/paper/2209.07330/) (2022)
- [Doubly Robust Estimation of Direct and Indirect Quantile Treatment Effects with Machine Learning](https://mlsanity.com/paper/2307.01049/) (2023)
- [Efficient and Sharp Off-Policy Evaluation in Robust Markov Decision Processes](https://mlsanity.com/paper/2404.00099/) (2024)
- [Assumption-lean falsification tests of rate double-robustness of double-machine-learning estimators](https://mlsanity.com/paper/2306.10590/) (2023)
- [Choice Models and Permutation Invariance: Demand Estimation in Differentiated Products Markets](https://mlsanity.com/paper/2307.07090/) (2023)
- [Automatic debiasing of neural networks via moment-constrained learning](https://mlsanity.com/paper/2409.19777/) (2024)
- [Detecting Instruction Fine-tuning Attacks using Influence Function](https://mlsanity.com/paper/2504.09026/) (2025)
- [Beyond the Average: Distributional Causal Inference under Imperfect Compliance](https://mlsanity.com/paper/2509.15594/) (2025)
- [Data-Driven Influence Functions for Optimization-Based Causal Inference](https://mlsanity.com/paper/2208.13701/) (2022)
- [The Implicit Delta Method](https://mlsanity.com/paper/2211.06457/) (2022)
- [SLEM: Machine Learning for Path Modeling and Causal Inference with Super Learner Equation Modeling](https://mlsanity.com/paper/2308.04365/) (2023)
- [Double Debiased Machine Learning for Mediation Analysis with Continuous Treatments](https://mlsanity.com/paper/2503.06156/) (2025)
- [Density Ratio-Free Doubly Robust Proxy Causal Learning](https://mlsanity.com/paper/2505.19807/) (2025)
- [Assessing the robustness of heterogeneous treatment effects in survival analysis under informative censoring](https://mlsanity.com/paper/2510.13397/) (2025)
- [Debiased neural operators for estimating functionals](https://mlsanity.com/paper/2604.19296/) (2026)
- [Debiased Regression for Root-N-Consistent Conditional Mean Estimation](https://mlsanity.com/paper/2411.11748/) (2024)

## Abstract (arXiv)

Many useful parameters depend on nonparametric first steps. Examples include games, dynamic discrete choice, average exact consumer surplus, and treatment effects. Often estimators of these parameters are asymptotically equivalent to a sample average of an object referred to as the influence function. The influence function is useful in local policy analysis, in evaluating local sensitivity of estimators, constructing debiased machine learning estimators, in efficiency comparisons, and in formulating primitive regularity conditions for asymptotic normality, We show that the influence function is a Gateaux derivative with respect to a smooth deviation evaluated at a point mass. This result generalizes the classic Von Mises (1947) and Hampel (1974) calculation to estimators that depend on smooth nonparametric first steps. We give explicit influence functions for first steps that satisfy exogenous or endogenous orthogonality conditions. We use these results to generalize the omitted variable bias formula for regression to policy analysis for and sensitivity to structural changes. We apply this analysis and find no sensitivity to endogeneity of average equivalent variation estimates in a gasoline demand application.