Group Permutation Testing in Linear Model: Sharp Validity, Power Improvement, and Extension Beyond Exchangeability

TL;DR

This work develops a group-permutation framework for finite-sample inference in the fixed-design linear model $Y=Z\beta+bX+\varepsilon$ to obtain sharp Type I error control and improved power under complex noise structures.Central ideas include extending PALMRT and CPT via a finite permutation group $\mathcal{P}_K$ to provide exact- or near-exact-calibration results, and introducing a design-dependent optimization to tailor the permutation group for power gains.Beyond exact exchangeability, the paper connects to conformal prediction, yielding robustness guarantees that degrade with total-variation distances between $\varepsilon$ and its group-permuted versions, and recovers exact validity when exchangeability holds.Together, the results offer a principled bridge from exact randomization validity to design-adaptive, robustness-aware inference in linear models, with practical implications for small-sample, high-correlation designs and heterogeneous noise.Overall, the group-permutation perspective provides a versatile toolkit for exact-style inference and its robust extensions in finite samples.

Abstract

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.

Figures (3)

• Figure 1: Type II Error for Gaussian X and $\epsilon$, a=0.1
• Figure 2: Type II Error for $t_2$ X and Gaussian $\epsilon$, a=0.1
• Figure 3: Probability density of $\Vert H^{Z}e_{i}\Vert^{2}_{2}$

Theorems & Definitions (33)

• Proposition 1
• Theorem 2
• Proposition 2
• Theorem 3: Upper and lower bound of $X^{T}H^{ZZ_{\pi}}X$
• Lemma 4
• Theorem 5: Informal
• Theorem 6: Lower bound of $\lambda_2$, informal
• Proposition 3
• Theorem 9
• Theorem 10