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.

翻译：我们针对固定设计线性模型 $Y = Zβ+ bX + \varepsilon$ 中单个回归系数的有限样本推断问题展开研究，其中 $\varepsilon\in\mathbb{R}^n$ 可能呈现复杂的相依性或异质性。通过构建群置换框架，我们为线性模型检验建立了统一且可分析的随机化结构。在可交换误差假设下，我们将置换增强回归检验置于该群论框架中，证明对任意置换群而言，分组版PALMRT方法可将第一类错误率控制在$2α$水平内；进一步通过最坏情形构造表明因子$2$具有精确性，且无需额外假设时无法改进。其次，我们将第二类错误与设计依赖的几何分离度关联，将其表述为关于置换群的组合优化问题，并在温和的次高斯假设下给出其界值。针对第二类错误上界控制，我们提出一种构造性算法，其置换策略效果优于（至少不劣于）独立同分布置换，仿真实验表明该方法在重尾设计下具有显著功效提升。最后，通过将基于秩的随机化论证与共形推断相结合，我们将基于群组的CPT和PALMRT方法推广至超出可交换性假设的情形。所得加权群检验满足有限样本第一类错误界，该界值随$\varepsilon$与其群置换版本间总变差距离的加权平均值呈优雅衰减：当这些差异消失时恢复精确有效性，否则提供定量鲁棒性保证。综上，群置换视角为从精确随机化有效性过渡到设计自适应功效和近似对称性下的定量鲁棒性建立了理论桥梁。