Adaptive nonparametric locally optimum Bayes detection in additive non-Gaussian noise

A nonparametric generalization of the locally optimum Bayes (LOB) parametric theory of signal detection in additive non-Gaussian noise with independent sampling is presented. From a locally asymptotically normal (LAN) expansion of the log-likelihood ratio the nonparametric detector structure, in both coherent and incoherent modes, is determined. Moreover, its statistics under both hypotheses are obtained. The nonparametric LAN log-likelihood ratio is then reduced to a least informative (i.e., having minimum variance under the hypothesis, H₀) local parametric submodel, which is referred to as adaptive. In the adaptive submodel, certain nonlinearities are replaced by their efficient estimates. This is accomplished such that no information is lost when the noise first-order density is no longer parametrically defined. Adaptive nonparametric LOB detectors are thus shown to be asymptotically optimum (AO), canonical in signal waveform, distribution free in noise statistics, and identical in form (in the symmetric cases) to their parametric counterparts. A numerical example is provided when the underlying density is Middleton´s (see ibid., vol.45, p.1129-49, May 1999)Class-A noise, which demonstrates that even with a relatively small sample size (O(10²)) adaptive LOB nonparametric detectors perform nearly as well as the classical LOB detectors.