Adaptive detection of known signals in additive noise by means of kernel density estimators

We consider the problem of detecting known signals contaminated by additive noise with a completely unknown probability density function f. To this end, we propose a new adaptive detection rule. It is defined by plugging a kernel density estimator fˆ of f into the maximum a posteriori (MAP) detector. The estimate fˆ can either be computed off-line from a training sequence or on-line simultaneously with the detection. For the off-line detector, we prove that the (asymptotic) error probability for weak signals converges to the minimal error probability of the MAP detector as the number of training data tends to infinity, and we also establish rates of convergence and the optimal choice of bandwidth order for a certain class of noise densities. In a Monte Carlo study, the off-line plug-in MAP detectors are compared with the L¹- and L²-detectors for various noise distributions. When the training sequence is long enough, the plug-in detectors have excellent performance for a wide range of distributions, whereas the L²-detector breaks down for heavy-tailed distributions and the L¹-detector for distributions with little mass around the origin