Two-stage Wilcoxon detectors using conditional tests

The idea of the Wilcoxon detector using a conditional test is extended to two-stage detectors, which constitute a class of sequential detectors. Three different two-stage detectors are considered, using the Wilcoxon statistic with conditional tests. The three detectors differ in the usage of the samples of the first stage for computing the test statistic in the second stage. In computing the test statistic in the second stage, the first detector disregards all the samples of the first stage, the second detector uses the value of the test statistic of the first stage, and the third detector uses all the samples of the first stage. The asymptotic relative efficiency (ARE) improves from the first to the second and third detectors. However, the ranking requirement is worst for the third detector, and best for the second detector. Comparison of these detectors with fixed-sample-size (FSS) detectors shows that they are better than both the FSS linear detector and the FSS conditional Wilcoxon detector under Gaussian noise, in terms of the average sample size under the same false-alarm and detection probabilities.