Asymptotic Normality of Log Likelihood Ratio and Fundamental Limit of the Weak Detection for Spiked Wigner Matrices
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910750013915136 |
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| author | Chung, Hye Won Lee, Jiho Lee, Ji Oon |
| author_facet | Chung, Hye Won Lee, Jiho Lee, Ji Oon |
| contents | We consider the problem of detecting the presence of a signal in a rank-one spiked Wigner model. For general non-Gaussian noise, assuming that the signal is drawn from the Rademacher prior, we prove that the log likelihood ratio (LR) of the spiked model against the null model converges to a Gaussian when the signal-to-noise ratio is below a certain threshold. The threshold is optimal in the sense that the reliable detection is possible by a transformed principal component analysis (PCA) above it. From the mean and the variance of the limiting Gaussian for the log-LR, we compute the limit of the sum of the Type-I error and the Type-II error of the likelihood ratio test. We also prove similar results for a rank-one spiked IID model where the noise is asymmetric but the signal is symmetric. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_00821 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Asymptotic Normality of Log Likelihood Ratio and Fundamental Limit of the Weak Detection for Spiked Wigner Matrices Chung, Hye Won Lee, Jiho Lee, Ji Oon Statistics Theory Probability Machine Learning 62H15, 60F05, 82B44 We consider the problem of detecting the presence of a signal in a rank-one spiked Wigner model. For general non-Gaussian noise, assuming that the signal is drawn from the Rademacher prior, we prove that the log likelihood ratio (LR) of the spiked model against the null model converges to a Gaussian when the signal-to-noise ratio is below a certain threshold. The threshold is optimal in the sense that the reliable detection is possible by a transformed principal component analysis (PCA) above it. From the mean and the variance of the limiting Gaussian for the log-LR, we compute the limit of the sum of the Type-I error and the Type-II error of the likelihood ratio test. We also prove similar results for a rank-one spiked IID model where the noise is asymmetric but the signal is symmetric. |
| title | Asymptotic Normality of Log Likelihood Ratio and Fundamental Limit of the Weak Detection for Spiked Wigner Matrices |
| topic | Statistics Theory Probability Machine Learning 62H15, 60F05, 82B44 |
| url | https://arxiv.org/abs/2203.00821 |