In evaluating the performance of detectors such as the orthogonal subspace projection (OSP) detector, it is often assumed that the model under which the detector is constructed is the correct model. However, in practice, the ability to identify all background endmembers might be limited. Consequently, the OSP detector would use only a subset of all background endmembers. In this paper, we consider consequences of using such incomplete information. To this end, we contrast between two detectors, one that uses all endmembers and another one that uses only some endmembers. The remaining endmembers might be unknown or deliberately discarded if we believe that this would improve the detector performance. We provide formulas for detection power of the two detectors. We also calculate their relative efficiency, which is a useful tool for comparison of detectors without directly calculating the detection power. We then show some theorems that help us in understanding relationships between the power detection of the two detectors. We also provide numerical results demonstrating the consequences of our theorems in specific scenarios. We show when the detector using more information is more powerful, but we also show examples of the opposite to be true. Practical consequences of using a given number of endmembers in the OSP detector are evaluated using an AVIRIS hyperspectral image. The results show that using too many endmembers is not beneficial for the OSP detector, and the optimal number of endmembers to be used depends on the size of the target and the desired false alarm level.
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