This paper tackles the problem of detecting a random signal embedded in additive white noise. Although the likelihood ratio test (LRT) is the well-known optimum detector for this problem, it may not be easily realized in applications such as radar, sonar, seismic, digital communications, speech analysis and automatic fault detection in machinery, for which suboptimal quadratic detectors have been extensively employed. In this paper, the relationships between four suboptimal quadratic detection schemes, namely, the energy, matched subspace, maximum deflection ratio as well as spectrum matching detectors, and the LRT are studied. In particular, we show that each of those suboptimal detectors can approach the optimal LRT under certain operating conditions. These results are verified via Monte Carlo simulations.