Weak Detection in the Spiked Wigner Model

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