April 2024 Minimax rates for heterogeneous causal effect estimation
Edward H. Kennedy, Sivaraman Balakrishnan, James M. Robins, Larry Wasserman
Author Affiliations +
Ann. Statist. 52(2): 793-816 (April 2024). DOI: 10.1214/24-AOS2369


Estimation of heterogeneous causal effects—that is, how effects of policies and treatments vary across subjects—is a fundamental task in causal inference. Many methods for estimating conditional average treatment effects (CATEs) have been proposed in recent years, but questions surrounding optimality have remained largely unanswered. In particular, a minimax theory of optimality has yet to be developed, with the minimax rate of convergence and construction of rate-optimal estimators remaining open problems. In this paper, we derive the minimax rate for CATE estimation, in a Hölder-smooth nonparametric model, and present a new local polynomial estimator, giving high-level conditions under which it is minimax optimal. Our minimax lower bound is derived via a localized version of the method of fuzzy hypotheses, combining lower bound constructions for nonparametric regression and functional estimation. Our proposed estimator can be viewed as a local polynomial R-Learner, based on a localized modification of higher-order influence function methods. The minimax rate we find exhibits several interesting features, including a nonstandard elbow phenomenon and an unusual interpolation between nonparametric regression and functional estimation rates. The latter quantifies how the CATE, as an estimand, can be viewed as a regression/functional hybrid.

Funding Statement

EK gratefully acknowledges support from NSF Grant DMS-1810979, NSF CAREER Award 2047444 and NIH R01 Grant LM013361-01A1, and SB and LW from NSF Grant DMS-1713003. EK also thanks Matteo Bonvini and Tiger Zeng for very helpful discussions.


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Edward H. Kennedy. Sivaraman Balakrishnan. James M. Robins. Larry Wasserman. "Minimax rates for heterogeneous causal effect estimation." Ann. Statist. 52 (2) 793 - 816, April 2024. https://doi.org/10.1214/24-AOS2369


Received: 1 March 2022; Revised: 1 December 2023; Published: April 2024
First available in Project Euclid: 9 May 2024

Digital Object Identifier: 10.1214/24-AOS2369

Primary: 62G08 , 62H12

Keywords: Causal inference , Functional estimation , higher-order influence functions , Nonparametric regression , Optimal rates of convergence

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 2 • April 2024
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