Abstract

Estimation of treatment effects is of great importance and has drawn much atten

tion in various areas. In the potential outcomes framework, since potential outcomes refer

to the same characteristic under different treatment assignments, they usually share certain

similarities. Such similarities are largely overlooked in the literature, leading to potential

efficiency loss. In this paper, we introduce a semi-parametric proportional likelihood ratio

model (SPLRM) to jointly model the conditional distributions of potential outcomes through

a shared baseline distribution, fully utilizing information from both treatment and control

groups. We estimate all underlying parameters and general causal estimands by maximum

empirical likelihood estimation. An iterative empirical likelihood algorithm is developed for

parameter estimation, and a simple likelihood ratio test is introduced to assess the distributional treatment effect. We show that the proposed estimators for various treatment effects

are asymptotically normal, and the likelihood ratio test statistic follows asymptotically a

central Chi-square distribution when there is no distributional treatment effect. This approach improves efficiency and robustness compared to traditional methods that separately

estimate treatment effects.

Simulations and an analysis of the National Supported Work

Demonstration dataset demonstrate the practical applicability and advantages of the proposed SPLRM-based method.

Key words and phrases: Causal inference, Distributional treatment effect, Likelihood ratio test, Semiparametric proportional likelihood ratio model

Information

Preprint No.SS-2025-0267
Manuscript IDSS-2025-0267
Complete AuthorsManli Cheng, Yukun Liu
Corresponding AuthorsYukun Liu
Emailsykliu@sfs.ecnu.edu.cn

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Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (12571283) and the Fundamental and Interdisciplinary Disciplines Breakthrough

Plan of the Ministry of Education of China (JYB2025XDXM904).

Yukun Liu is the

Supplementary Materials

The supplementary material contains the proofs of Theorems 1–4 and Corollary 1,

additional simulation results, and supplementary results for the real data analysis.

Supplementary materials are available for download.