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arXiv · 2607.02295

MATCH: Multiplier-Assisted Tests for Conditional Hypotheses in Non-Euclidean Data

Leheng Cai·Xu Guo·Qirui Hu
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Abstract

We propose a new procedure MATCH (Multiplier-Assisted Tests for Conditional Hypotheses) to test whether the non-Euclidean data match the target model, which is a general framework for significance and specification testing in Fr\'echet regression. MATCH covers global significance, partial significance, and the adequacy of global Fr\'echet regression, providing a unified way to compare unrestricted conditional Fr\'echet means with restricted alternatives. One of the key challenges is that the ordinary held-out loss difference is first-order degenerate under the null: the oracle losses coincide, and plug-in statistics is dominated by nuisance estimation error. MATCH uses sample splitting and independent random multipliers on held-out losses to create a nondegenerate Gaussian leading term without residuals or tangent-space coordinates. To improve data use and stability, we further develop cross-fitted tests and repeated cross-fitting with p-value merging. We establish asymptotic null validity, consistency under fixed alternatives, and local power guarantees. Simulations for distributional, symmetric positive-definite (SPD) matrix-valued, and spherical responses support the theoretical findings, and applications to county-level household income distributions and North Atlantic tropical-cyclone locations demonstrate the practical use of the proposed tests.

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Leheng Cai, Xu Guo, Qirui Hu. 2026-07-02. MATCH: Multiplier-Assisted Tests for Conditional Hypotheses in Non-Euclidean Data. https://arxiv.org/abs/2607.02295

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