Minimax rates for pointwise CATE estimation: the role of design smoothness

Yuxuan Hou; Zhenghan Li

math.STOct 13, 2026

Minimax rates for pointwise CATE estimation: the role of design smoothness

Authors: Yuxuan Hou, Zhenghan Li

Organizations: Qiuzhen College, Tsinghua University · School of Statistics, University of Minnesota

Abstract

We study the role of design smoothness in the minimax estimation of conditional average treatment effects at a fixed covariate point. For design densities bounded above and away from zero, without further regularity, we derive upper and lower risk bounds with an arbitrarily small gap in the convergence-rate exponent. Under uniform design, we obtain matching upper and lower bounds at interior points in the regime of low nuisance smoothness. These results show that rough covariate designs can lead to slower rates than uniform designs, even under the same smoothness assumptions on the treatment effect and nuisance functions, highlighting the role of the covariate distribution in pointwise causal estimation.

Conditional Average Treatment Effect Estimation

Conditional average treatment effect estimation measures how treatment effects vary with observed characteristics to guide targeted decisions.

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Sharp Bounds on Continuous Treatment Effects under Unobserved Confounding

Junzhe Zhang

Syracuse University

Causal Effect Estimation Causal Inference with Unobserved Confounding Conditional Average Treatment Effect Estimation Causal Identifiability

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We study partial identification of conditional treatment effects with a continuous treatment under unobserved confounding. We target the conditional average potential outcome (CAPO) E[Y_x|z] and the conditional average treatment effect (CATE) between two treatment levels. We assume a bounded outcome and Lipschitz continuity of the response functions and of the propensity scores given the latent confounder, and show that these conditions imply the smooth treatment response and smooth treatment selection conditions of Kim et al. (2018). The combined bound of Kim et al. uses only two reference points on the identified diagonal and is generally loose. Bounding through every diagonal reference point instead gives explicit bounds whose average is the sharp identified set of the CAPO at every treatment, inside or outside the observational support. Capping the difference between the upper and lower bounds gives sharp causal bounds on the CATE. These bounds are informative without monotonicity or overlap assumptions.

Abstract

Conditional Average Treatment Effect Estimation

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