Daisuke Kurisu, Yuta Okamoto and Taisuke Otsu

Published 22 January 2026

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In applied research, Lee (2009) bounds are widely applied to bound the average treat- ment effect in the presence of selection bias. This paper extends the methodology of Lee bounds to accommodate outcomes in a general metric space, such as compositional and distributional data. By exploiting a representation of the Frechet mean of the potential outcome via embedding in an Euclidean or Hilbert space, we present a feasible characterization of the identified set of the causal effect of interest, and then propose its analog estimator and bootstrap confidence region. The proposed method is illustrated by numerical examples on compositional and distributional data.