On the asmptotic efficiency of locally optimum detectors

A detector examines an unknown waveform to determine whether it is a mixture of signal and noise, or noise alone. The Neyman-Pearson detector is optimum in the sense that for given false alarm probability, signal-to-noise ratio, and number of observations, it minimizes the false dismissal probability. This detector is optimum for all values of the signal-to-noise ratio, and its implementation is usually quite complicated. In many situations it is desired to detect signals which are very weak compared to the noise. The locally optimum detector is defined as one which has optimum properties only for small signal-to-noise ratios. It is proposed as an alternative to the Neyman-Pearson detector, since in practice it is usually only necessary to have a near-optimum detector for weak signals, since strong signals will be detected with reasonable accuracy even if the detector is well below optimum. In order to evaluate the performance of the locally optimum detector, it is compared to the Neyman-Pearson detector. This comparison is based on the concept of asymptotic relative efficiency introduced by Pitman for comparing hypothesis testing procedures. On the basis of this comparison, it is shown that the locally optimum detector is asymptotically as efficient as the Neyman-Pearson detector. A number of applications to several detection problems are considered. It is found that the implementation of the locally optimum detector is less, or at most as complicated as that of the Neyman-Pearson detector.