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The optimal receiver operating characteristic (ROC) curve, giving the maximumprobability of detection as a function of the probability of false alarm, is akey information-theoretic indicator of the difficulty of a binary hypothesistesting problem (BHT). It is well known that the optimal ROC curve for a givenBHT, corresponding to the likelihood ratio test, is theoretically determined bythe probability distribution of the observed data under each of the twohypotheses. In some cases, these two distributions may be unknown orcomputationally intractable, but independent samples of the likelihood ratiocan be observed. This raises the problem of estimating the optimal ROC for aBHT from such samples. The maximum likelihood estimator of the optimal ROCcurve is derived, and it is shown to converge to the true optimal ROC curve inthe \levy\ metric, as the number of observations tends to infinity. A classicalempirical estimator, based on estimating the two types of error probabilitiesfrom two separate sets of samples, is also considered. The maximum likelihoodestimator is observed in simulation experiments to be considerably moreaccurate than the empirical estimator, especially when the number of samplesobtained under one of the two hypotheses is small. The area under the maximumlikelihood estimator is derived; it is a consistent estimator of the true areaunder the optimal ROC curve.

Maximum Likelihood Estimation of Optimal Receiver Operating Characteristic Curves from Likelihood Ratio Observations