Asymptotic Normality of Log Likelihood Ratio and Fundamental Limit of the Weak Detection for Spiked Wigner Matrices

• URL: http://arxiv.org/abs/2203.00821v1
• Date: Wed, 2 Mar 2022 02:14:54 GMT
• Title: Asymptotic Normality of Log Likelihood Ratio and Fundamental Limit of the Weak Detection for Spiked Wigner Matrices
• Authors: Hye Won Chung, Jiho Lee, Ji Oon Lee
• Abstract summary: We consider the problem of detecting the presence of a signal in a rank-one spiked Wigner model. We prove that the log likelihood ratio of the spiked model against the null model converges to a Gaussian when the signal-to-noise ratio is below a certain threshold.
• Score: 13.653940190782142
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the problem of detecting the presence of a signal in a rank-one spiked Wigner model. Assuming that the signal is drawn from the Rademacher prior, we prove that the log likelihood ratio of the spiked model against the null model converges to a Gaussian when the signal-to-noise ratio is below a certain threshold. From the mean and the variance of the limiting Gaussian, we also compute the limit of the sum of the Type-I error and the Type-II error of the likelihood ratio test.

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