We derive approximate expressions for the probability of error in a two-class hypothesis testing problem in which the two hypotheses are characterized by zero-mean complex Gaussian distributions. These error expressions are given in terms of the moments of the test statistic employed and we derive these moments for both the likelihood ratio test, appropriate when class densities are known, and the generalized likelihood ratio test, appropriate when class densities must be estimated from training data. These moments are functions of class distribution parameters which are generally unknown so we develop unbiased moment estimators in terms of the training data. With these, accurate estimates of probability of error can be calculated quickly for both the optimal and plug-in rules from available training data. We present a detailed example of the behavior of these estimators and demonstrate their application to common pattern recognition problems, which include quantifying the incremental value of larger training data collections, evaluating relative geometry in data fusion from multiple sensors, and selecting a good subset of available features.