Weak Detection in the Spiked Wigner Model

We consider the weak detection problem in a rank-one spiked Wigner data matrix where the signal-to-noise ratio is small so that reliable detection is impossible. We prove a central limit theorem for the linear spectral statistics of general rank-one spiked Wigner matrices, and based on the central limit theorem, we propose a hypothesis test on the presence of the signal by utilizing the linear spectral statistics of the data matrix. The test is data-driven and does not require prior knowledge about the distribution of the signal or the noise. When the noise is Gaussian, the proposed test is optimal in the sense that its error matches that of the likelihood ratio test, which minimizes the sum of the Type-I and Type-II errors. If the density of the noise is known and non-Gaussian, the error of the test can be lowered by applying an entrywise transformation to the data matrix.