Monotone Response for Random Objects

Monotone treatment response (MTR), monotone treatment selection (MTS), and monotone instrumental variable (MIV) assumptions are widely used to partially identify counterfactual mean outcomes, but existing analyses have focused almost exclusively on scalar outcomes. We develop a unified framework for partial identification with outcomes that take values in a general metric space under these monotonicity restrictions by embedding the metric space into an $L^2$ space and imposing coordinatewise monotonicity on the embedded functions. The proposed framework yields valid identified sets for Fr\'echet means in a broad class of random-object spaces and further delivers sharp identification results for distributional outcomes under the Wasserstein metric, interval-valued outcomes represented by support functions, and compositional outcomes under the Aitchison metric. We also establish a support-free characterization of the identified set under the joint MTR--MTS assumption. Numerical and empirical illustrations based on Job Corps earnings data and periodontal health distributions from the National Health and Nutrition Examination Survey demonstrate the empirical usefulness of the proposed framework.