Decision functions for many common detection/discrimination tasks can be cast in the form of functions which are linear in the image data. This is the case, for example, for the simple matched filter result in the case of detection of a known signal in known location against an additive Gaussian noise background. For other tasks the ideal (maximum likelihood) observer decision rule is higher order in the data. One example of this is detection of a known signal as above but with position unknown. Wagner and Barrett have examined this subject in some detail and have suggested that the quadratic term in such higher order decision tasks may be of primary importance.' We have simulated a variety of position unknown detection tasks in order to calculate the efficiency (vis-a-vis the exact maximum likelihood decision rule) of using the first, second, or higher order terms. We find that the second order term may dominate in some cases, but that for others it is inferior by a factor of 2 or more (especially for low contrast, non-zero mean objects) in comparison with the ideal observer decision function.