Neyman-Orthogonal Tests for Equality of Regression Functions

We develop a test for equality of regression functions across two populations that allows unequal covariate distributions and uses flexible modern learners for regression estimation. We target the regression contrast through pseudo-residuals, which we construct by multiplying residuals from the pooled outcome regression by the difference between the population indicator and its propensity. Under overlap, the conditional mean of the proposed pseudo-residuals identifies the regression contrast. Furthermore, the pseudo-residuals satisfy Neyman orthogonality, which removes the first-order population bias from nuisance estimation and allows us to test the regression contrast under weaker nuisance estimation rate requirements. To construct our test statistic, we embed the conditional moment of the pseudo-residuals in a reproducing kernel Hilbert space (RKHS), leading to a simple closed form expression and calibrate its empirical norm using a Gaussian multiplier bootstrap. We establish a finite-sample calibration bound that separates oracle Gaussian approximation, nuisance estimation, and covariance estimation errors. Our main result gives conditions under which we can use the same observations for both nuisance fitting and testing, yielding valid and powerful inference without sample splitting. In a fixed Gaussian regression model, we establish matching lower and upper bounds for the minimax separation order (n^{-1/2}) in the RKHS discrepancy and show that the proposed test attains this order. We conduct extensive numerical studies to examine size and power across learners and demonstrate power gains over a single sample split and other competitors and illustrate the test on an exploratory application to negotiation dialogues for regression comparison with text covariates.