LTCC: Localized Testing of Coordinate Contributions to Conditional Distribution Discrepancy

We study whether a prespecified response coordinate contributes to the discrepancy between two multivariate conditional distributions at a fixed covariate profile. We define a localized no-contribution hypothesis that requires preservation of the target coordinate's conditional marginal distribution and its marginal-standardized cross-association with the remaining responses. Population-specific conditional probability integral transforms separate association changes from marginal differences in other coordinates. Characteristic kernels yield an equivalent direct-sum reproducing kernel Hilbert space characterization. We derive an exact von Mises expansion accounting for estimated conditional distributions and construct a cross-fitted one-step estimator. We establish a Gaussian limit under fixed alternatives and a weighted-${\chi }^2$ limit under the null for the squared-norm statistic. A consistent multiplier bootstrap yields asymptotic level $\alpha$ and consistency against fixed alternatives. Simulations and applications to large language models and image classification illustrate the proposed testing procedure.