Group Permutation Testing in Linear Model: Sharp Validity, Power Improvement, and Extension Beyond Exchangeability
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| Format: | Preprint |
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2026
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| _version_ | 1866917415144652800 |
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| author | Li, Zonghan Zhou, Hongyi Zhang, Zhiheng |
| author_facet | Li, Zonghan Zhou, Hongyi Zhang, Zhiheng |
| contents | We consider finite-sample inference for a single regression coefficient in the fixed-design linear model $Y = Zβ+ bX + \varepsilon$, where $\varepsilon\in\mathbb{R}^n$ may exhibit complex dependence or heterogeneity. We develop a group permutation framework, yielding a unified and analyzable randomization structure for linear-model testing. Under exchangeable errors, we place permutation-augmented regression tests within this group-theoretic setting and show that a grouped version of PALMRT controls Type I error at level at most $2α$ for any permutation group; moreover, we provide an worst-case construction demonstrating that the factor $2$ is sharp and cannot be improved without additional assumptions. Second, we relate the Type II error to a design-dependent geometric separation. We formulate it as a combinatorial optimization problem over permutation groups and bound it under additional mild sub-Gaussian assumptions. For the Type II error upper bound control, we propose a constructive algorithm for the permutation strategy that is better (at least no worse) than the i.i.d. permutation, with simulations empirically indicating substantial power gains, especially under heavy-tailed designs. Finally, we extend group-based CPT and PALMRT beyond exchangeability by connecting rank-based randomization arguments to conformal inference. The resulting weighted group tests satisfy finite-sample Type I error bounds that degrade gracefully with a weighted average of total variation distances between $\varepsilon$ and its group-permuted versions, recovering exact validity when these discrepancies vanish and yielding quantitative robustness otherwise. Taken together, the group-permutation viewpoint provides a principled bridge from exact randomization validity to design-adaptive power and quantitative robustness under approximate symmetries. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_17734 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Group Permutation Testing in Linear Model: Sharp Validity, Power Improvement, and Extension Beyond Exchangeability Li, Zonghan Zhou, Hongyi Zhang, Zhiheng Methodology 62J05 (Primary) 62G10, 62E15 (Secondary) We consider finite-sample inference for a single regression coefficient in the fixed-design linear model $Y = Zβ+ bX + \varepsilon$, where $\varepsilon\in\mathbb{R}^n$ may exhibit complex dependence or heterogeneity. We develop a group permutation framework, yielding a unified and analyzable randomization structure for linear-model testing. Under exchangeable errors, we place permutation-augmented regression tests within this group-theoretic setting and show that a grouped version of PALMRT controls Type I error at level at most $2α$ for any permutation group; moreover, we provide an worst-case construction demonstrating that the factor $2$ is sharp and cannot be improved without additional assumptions. Second, we relate the Type II error to a design-dependent geometric separation. We formulate it as a combinatorial optimization problem over permutation groups and bound it under additional mild sub-Gaussian assumptions. For the Type II error upper bound control, we propose a constructive algorithm for the permutation strategy that is better (at least no worse) than the i.i.d. permutation, with simulations empirically indicating substantial power gains, especially under heavy-tailed designs. Finally, we extend group-based CPT and PALMRT beyond exchangeability by connecting rank-based randomization arguments to conformal inference. The resulting weighted group tests satisfy finite-sample Type I error bounds that degrade gracefully with a weighted average of total variation distances between $\varepsilon$ and its group-permuted versions, recovering exact validity when these discrepancies vanish and yielding quantitative robustness otherwise. Taken together, the group-permutation viewpoint provides a principled bridge from exact randomization validity to design-adaptive power and quantitative robustness under approximate symmetries. |
| title | Group Permutation Testing in Linear Model: Sharp Validity, Power Improvement, and Extension Beyond Exchangeability |
| topic | Methodology 62J05 (Primary) 62G10, 62E15 (Secondary) |
| url | https://arxiv.org/abs/2601.17734 |