Locally optimum nonparametric methods for the detection of signals in noise

Abstract

The Neyman-Pearson detector and other decision-theoretic detectors are designed for a particular distribution function. This distribution function can be indexed by a finite, or at most a countable number of real parameters. Thus, these detectors may be classified as parametric detection systems. In the present work a nonparametric detection scheme is discussed. This detector is designed for a wide class of distribution functions which can not be indexed by a finite, or countable number of real parameters. The operation of the nonparametric detector depends on the assumption that a control-sample of noise is available. The presence of the signal is determined by comparing the unknown waveform with the control-sample. If attention is restricted to the class of nonparametric tests known as rank tests, it is possible to design an optimum nonparametric detector, in the sense that it is based on the optimum rank test. In general, the structure of the optimum nonparametric detector is quite complicated. However, in the practically important case of threshold detection, the locally optimum nonparametric detector assumes a particularly simple form. The concept of asymptotic relative efficiency is used to compare the locally optimum nonparametric detector with the Neyman-Pearson detector. It is shown that, on the basis of this comparison, the locally optimum nonparametric detector is asymptotically as efficient as the Neyman-Pearson detector. A number of applications to specific detection problems is considered. In the problem of detecting a constant signal in additive noise, it is shown that the locally optimum non-parametric detector is less sensitive to the functional form of the noise distribution than is the Neyman-Pearson detector.