Approximating the test statistic distribution and ALROC in signal-detection tasks with signal location uncertainty

In medical imaging, signal detection is one of the most important tasks. It is especially important to study detection tasks with signal location uncertainty. One way to evaluate system performance on such tasks is to compute the area under the localization-receiver operating characteristic (LROC) curve. In an LROC study, detecting a signal includes two steps. The first step is to compute a test statistic to determine whether the signal is present or absent. If the signal is present, the second step is to identify the location of the signal. We use the test statistic which maximizes the area under the LROC curve (ALROC). We attempt to capture the distribution of this ideal LROC test statistic with signal-absent data using the extreme value distribution. Some simulated test statistics are shown along with extreme value distributions to illustrate how well our approximation captures the characteristics of the ideal LROC test statistic. We further derive an approximation to the ideal ALROC using the extreme value distribution and compare it to the direct simulation of the ALROC. Using a different approach by defining a parameterized probability density function of the data, we are able to derive another approximation to the ideal ALROC for weak signals from a power series expansion in signal amplitude.