Weak detection of signal in the spiked Wigner model

We consider the problem of detecting the presence of the signal in a rank-one signal-plus-noise data matrix. In case the signal-to-noise ratio is under a certain threshold, we propose a hypothesis testing based on the linear spectral statistics of the data matrix. The error of the proposed test matches that of the likelihood ratio test, which minimizes the sum of the Type-I and Type-II errors. The test does not depend on the distribution of the signal or the noise, and it can be extended to an adaptive test, which does not require the knowledge on the value of the signal-to-noise ratio, but performs better than random guess. As an intermediate step, we establish a central limit theorem for the linear spectral statistics of rank-one spiked Wigner matrices for a general spike.