Finite-Sample Inference for Sparsely Permuted Linear Regression

TL;DR

This paper develops a finite-sample statistical inference framework for permuted linear regression, addressing the challenge of unknown permutations in data matching, with methods validated through simulations and real data application.

Contribution

It introduces a localization step to reduce permutation complexity and provides valid inference procedures for permutation and regression coefficients in permuted linear models.

Findings

Finite-sample Type-I error control for permutation tests

High power to detect data mismatches

Scalable polynomial-time algorithm for permutation inference

Abstract

We study a linear observation model with an unknown permutation called \textit{permuted/shuffled linear regression}, where responses and covariates are mismatched and the permutation forms a discrete, factorial-size parameter. The permutation is a key component of the data-generating process, yet its statistical investigation remains challenging due to its discrete nature. We develop a general statistical inference framework on the permutation and regression coefficients. First, we introduce a localization step that reduces the permutation space to a small candidate set building on recent advances in the repro samples method, whose miscoverage decays polynomially with the number of Monte Carlo samples. Then, based on this localized set, we provide statistical inference procedures: a conditional Monte Carlo test of permutation structures with valid finite-sample Type-I error control. We also develop coefficient inference that remains valid under alignment uncertainty of permutations. For computational purposes, we develop a linear assignment problem computable in polynomial time and demonstrate that, with high probability, the solution is equivalent to that of the conventional least squares with large computational cost. Extensions to partially permuted designs and ridge regularization are further discussed. Extensive simulations and an application to air-quality data corroborate finite-sample validity, strong power to detect mismatches, and practical scalability.