The Influence Function of Transport-based Quantiles

Transport-based quantiles extend univariate quantiles to multivariate distributions via optimal transport. We study the influence function of the transport quantile map QP, defined as the optimal transport map pushing a fixed reference measure μ forward to a target distribution P. For the Huber contamination Pt=(1-t)P+tδx0, we prove that the first-order limit I(x0;QP(z)) := limtdownarrow 0 [Q_(1-t)P+tδx0(z)-QP(z)]/t exists whenever x0ne QP(z) and characterize it uniquely. Specifically, I(x0;QP(z))=nabla Gx0(z), where Gx0 is characterized by a uniformly elliptic equation with a Dirac source and a Neumann boundary condition. In every dimension d≥ 2, this influence function has a pole-type singularity. For fixed z∈operatornameint(Omegaμ), it remains bounded when FP(x0) stays away from z, where FP=QP-1 is the transport-based distribution function, but diverges as x0→QP(z), equivalently as FP(x0)→ z. In fact, |I(x0;QP(z))|asymp|z-FP(x0)|-(d-1). This contrasts with the bounded influence function of univariate quantiles and implies that I(X;QP(z)), for X~ P, has infinite second moment. Numerical experiments further suggest that empirical transport quantiles may exhibit stable-type non-Gaussian fluctuations.