Estimation ROC curves and their corresponding ideal observers

The LROC curve may be generalized in two ways. We can replace the location of the signal with an arbitrary set of parameters that we wish to estimate. We can also replace the binary function that determines whether an estimate is correct by a utility function that measures the usefulness of a particular estimate given the true parameter set. The expected utility for the true-positive detections may then be plotted versus the false-positive fraction as the detection threshold is varied to generate an estimation ROC curve (EROC). Suppose we run a 2AFC study where the observer must decide which image has the signal and then estimate the parameter set. Then the average value of the utility on those image pairs where the observer chooses the correct image is an estimate of the area under the EROC curve (AEROC). The ideal LROC observer may also be generalized to the ideal EROC observer, whose EROC curve lies above those of all other observers. When the utility function is non-negative the ideal EROC observer shares many properties with the ideal ROC observer, which can simplify the calculation of the ideal AEROC. When the utility function is concave the ideal EROC observer makes use of the posterior mean estimator. Other estimators that arise as special cases include maximum a posteriori estimators and maximum likelihood estimators. Multiple signals may be accomodated in this framework by making the number of signals one of the parameters in the set to be estimated.