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

Anna Bykhovskaya; Vadim Gorin; Sasha Sodin

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arXiv · 2507.18554

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

Anna Bykhovskaya·Vadim Gorin·Sasha Sodin

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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.

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Anna Bykhovskaya, Vadim Gorin, Sasha Sodin. 2025-07-24. How weak are weak factors? Uniform inference for signal strength in signal plus noise models. https://arxiv.org/abs/2507.18554

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