Testing relevant difference in high-dimensional linear regression with applications to detect transferability

2025/11/19 by Liu, Xu
Engineering · Mathematics · #FOS: Computer and information sciences #Methodology (stat.ME) #Random Matrices and Applications #Sparse and Compressive Sensing Techniques #Statistical Methods and Inference

paper · doi:10.48550/arxiv.2511.15236

openalex publication_date 2025/11/19 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28

Abstract

Most of researchers on testing a significance of coefficient \ubeta in high-dimensional linear regression models consider the classical hypothesis testing problem H₀^c: \ubeta=\uzero versus H₁^c: \ubeta ≠ \uzero. We take a different perspective and study the testing problem with the null hypothesis of no relevant difference between \ubeta and \uzero, that is, H₀: ‖\ubeta‖≤ δ₀ versus H₁: ‖\ubeta‖> δ₀, where δ₀ is a prespecified small constant. This testing problem is motivated by the urgent requirement to detect the transferability of source data in the transfer learning framework. We propose a novel test procedure incorporating the estimation of the largest eigenvalue of a high-dimensional covariance matrix with the assistance of the random matrix theory. In the more challenging setting in the presence of high-dimensional nuisance parameters, we establish the asymptotic normality for the proposed test statistics under both the null and alternative hypotheses. By applying the proposed test approaches to detect the transferability of source data, the unified transfer learning models simultaneously achieve lower estimation and prediction errors with comparison to existing methods. We study the finite-sample properties of the new test by means of simulation studies and illustrate its performance by analyzing the GTEx data.

Citations

• Unified Transfer Learning Models in High-Dimensional Linear Regression
• Score function-based tests for ultrahigh-dimensional linear models
• Testing for practically significant dependencies in high dimensions via bootstrapping maxima of U-statistics
• Statistical inference for function-on-function linear regression
• Transfer Learning under High-dimensional Generalized Linear Models
• Transfer Learning in Large-scale Gaussian Graphical Models with False Discovery Rate Control
• On the Theory of Transfer Learning: The Importance of Task Diversity
• Transfer Learning for High-dimensional Linear Regression: Prediction, Estimation, and Minimax Optimality
• Are deviations in a gradually varying mean relevant? A testing approach based on sup-norm estimators
• Integrative Factor Regression and Its Inference for Multimodal Data Analysis
• A Comprehensive Survey on Transfer Learning
• A Comprehensive Survey on Transfer Learning
• Transfer Learning for Nonparametric Classification: Minimax Rate and Adaptive Classifier
• Testing relevant hypotheses in functional time series via self-normalization
• Functional data analysis in the Banach space of continuous functions
• Conditional Mean and Quantile Dependence Testing in High Dimension
• Spectrum Estimation from Samples
• A General Theory of Hypothesis Tests and Confidence Regions for Sparse High Dimensional Models
• A general theory of hypothesis tests and confidence regions for sparse high dimensional models
• Detecting relevant changes in time series models
• Tests for High Dimensional Generalized Linear Models
• Spectrum estimation for large dimensional covariance matrices using random matrix theory

Related

• Transfer Learning for Linear Regression: a Statistical Test of Gain
• Testing Sufficiency for Transfer Learning
• Hypothesis Testing for the Covariance Matrix in High-Dimensional\n Transposable Data with Kronecker Product Dependence Structure
• Tests for High Dimensional Generalized Linear Models
• Hypothesis Testing for High-Dimensional Matrix-Valued Data