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The Influence Function of Semiparametric Estimators

arXiv 1508.01378 ↗First posted 6 Aug 2015stat.MEcited by 36

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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.

Key facts. Cited by 36 papers in the map (data through 7 Oct 2026). First posted on arXiv 6 Aug 2015.

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Citing papers per year: 2021: 2; 2022: 7; 2023: 6; 2024: 8; 2025: 9; 2026: 4.

36 papers.

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  1. 2022Robust and Agnostic Learning of Conditional Distributional Treatment Effectscited by 18
  2. 2022Inference on Strongly Identified Functionals of Weakly Identified Functionscited by 12
  3. 2024Simplifying debiased inference via automatic differentiation and probabilistic programmingcited by 10
  4. 2021A Simple and General Debiased Machine Learning Theorem with Finite Sample Guaranteescited by 9
  5. 2025On Efficient Estimation of Distributional Treatment Effects under Covariate-Adaptive Randomizationcited by 6
  6. 2023Adaptive Instrument Design for Indirect Experimentscited by 5
  7. 2024Learning Decision Policies with Instrumental Variables through Double Machine Learningcited by 5
  8. 2024Estimating Distributional Treatment Effects in Randomized Experiments: Machine Learning for Variance Reductioncited by 5
  9. 2025Automatic Debiased Machine Learning for Smooth Functionals of Nonparametric M-Estimandscited by 5
  10. 2021Multiply Robust Causal Mediation Analysis with Continuous Treatmentscited by 4
  11. 2022Optimal Best Arm Identification in Two-Armed Bandits with a Fixed Budget under a Small Gapcited by 4
  12. 2022A Free Lunch with Influence Functions? Improving Neural Network Estimates with Concepts from Semiparametric Statisticscited by 4
24 more
  1. 2023Doubly Robust Proximal Causal Learning for Continuous Treatmentscited by 4
  2. 2024Automated Efficient Estimation using Monte Carlo Efficient Influence Functionscited by 4
  3. 2022Best Arm Identification with Contextual Information under a Small Gapcited by 3
  4. 2023Doubly Robust Estimation of Direct and Indirect Quantile Treatment Effects with Machine Learningcited by 3
  5. 2024Efficient and Sharp Off-Policy Evaluation in Robust Markov Decision Processescited by 3
  6. 2023Assumption-lean falsification tests of rate double-robustness of double-machine-learning estimatorscited by 2
  7. 2024Automatic debiasing of neural networks via moment-constrained learningcited by 2
  8. 2025Beyond the Average: Distributional Causal Inference under Imperfect Compliancecited by 2
  9. 2022Data-Driven Influence Functions for Optimization-Based Causal Inferencecited by 1
  10. 2022The Implicit Delta Methodcited by 1
  11. 2023SLEM: Machine Learning for Path Modeling and Causal Inference with Super Learner Equation Modelingcited by 1
  12. 2025Double Debiased Machine Learning for Mediation Analysis with Continuous Treatmentscited by 1
  13. 2025Detecting Instruction Fine-tuning Attacks using Influence Functioncited by 1
  14. 2025Density Ratio-Free Doubly Robust Proxy Causal Learningcited by 1
  15. 2025Assessing the robustness of heterogeneous treatment effects in survival analysis under informative censoringcited by 1
  16. 2026Debiased neural operators for estimating functionalscited by 1
  17. 2023Choice Models and Permutation Invariance: Demand Estimation in Differentiated Products Marketscited by 0
  18. 2024Debiased Regression for Root-N-Consistent Conditional Mean Estimationcited by 0
  19. 2024Debiased Nonparametric Regression for Statistical Inference and Distributionally Robustnesscited by 0
  20. 2025Sample Complexity of Nonparametric Closeness Testing for Continuous Distributions and Its Application to Causal Discovery with Hidden Confoundingcited by 0
  21. 2025A Cautionary Tale on Integrating Studies with Disparate Outcome Measures for Causal Inferencecited by 0
  22. 2026Nonparametric Instrumental Variable Analysis Without Structural Equations: Debiased Inference on Functionals of Inverse Problems with No Solutionscited by 0
  23. 2026Modeling Covariate Transition for Efficient Estimation of Longitudinal Treatment Effects in Randomized Experimentscited by 0
  24. 2026Kernel-Based Functional Balancing for Causal Inference with Compositional Treatmentscited by 0