Taisuke Otsu and Kazuhiko Shinoda

Published 2 November 2026

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Mean effects do not fully describe how interventions change the distribution of an outcome. Counterfactual densities help characterize features such as concentration and asymmetry. When treatment varies continuously, interest often concerns how the outcome distribution changes across treatment levels. In a nonparametric model, however, inference at a specified treatment value generally requires smoothing over nearby treatment values. We define a finite-dimensional global projection of the counterfactual density onto a working family that may be misspecified. The projection minimizes a density discrepancy averaged over the marginal treatment distribution, avoiding the need for smoothing over nearby treatment values while allowing the approximating density to vary with treatment. We derive the efficient influence function for the projection parameter and establish asymptotic normality at the parametric rate for a bias-corrected estimator under empirical process and remainder conditions. Because the resulting approximation depends on the working family, we also develop inference for the difference between the optimized approximation risks of two non-nested families and construct a test of equal risks. Simulations examine the finite sample behavior of the projection estimator and the size and power of the test. An empirical illustration applies both procedures to blood lead and urinary albumin-creatinine ratio measurements from the National Health and Nutrition Examination Survey.