How weak are weak factors? Uniform inference for signal strength in signal plus noise models

2025-07-24

AI Summary

This paper addresses unified inference on signal strength in signal-plus-noise models—including factor models, spiked covariance matrices, low-rank perturbations of Wigner matrices, and canonical correlation analysis—by proposing a confidence interval construction method based on the *transitional distribution*. Departing from conventional piecewise Gaussian approximations that rely on ad hoc signal-strength regimes, our approach achieves uniformly valid inference across strong, weak, and critical signal regimes, thereby resolving the long-standing failure of inference in the critical regime. Theoretically grounded in random matrix theory and asymptotic statistics, the method models phase-transition behavior in the limiting spectral distribution of eigenvalues. Empirically, the procedure robustly identifies and quantifies weak-factor signals in macroeconomic and financial applications, demonstrating cross-model consistency and practical utility.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Graphical Models
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

The paper analyzes four classical signal-plus-noise models: the factor model, spiked sample covariance matrices, the sum of a Wigner matrix and a low-rank perturbation, and canonical correlation analysis with low-rank dependencies. The objective is to construct confidence intervals for the signal strength that are uniformly valid across all regimes - strong, weak, and critical signals. We demonstrate that traditional Gaussian approximations fail in the critical regime. Instead, we introduce a universal transitional distribution that enables valid inference across the entire spectrum of signal strengths. The approach is illustrated through applications in macroeconomics and finance.

Problem

• Construct uniform confidence intervals for signal strength
• Address failure of Gaussian approximations in critical regime
• Introduce universal transitional distribution for valid inference

Innovation

• Uniform inference for signal strength
• Universal transitional distribution introduced
• Valid across strong weak critical signals