  The optimal receiver operating characteristic (ROC) curve, giving the maximum
probability of detection as a function of the probability of false alarm, is a
key information-theoretic indicator of the difficulty of a binary hypothesis
testing problem (BHT). It is well known that the optimal ROC curve for a given
BHT, corresponding to the likelihood ratio test, is theoretically determined by
the probability distribution of the observed data under each of the two
hypotheses. In some cases, these two distributions may be unknown or
computationally intractable, but independent samples of the likelihood ratio
can be observed. This raises the problem of estimating the optimal ROC for a
BHT from such samples. The maximum likelihood estimator of the optimal ROC
curve is derived, and it is shown to converge to the true optimal ROC curve in
the \levy\ metric, as the number of observations tends to infinity. A classical
empirical estimator, based on estimating the two types of error probabilities
from two separate sets of samples, is also considered. The maximum likelihood
estimator is observed in simulation experiments to be considerably more
accurate than the empirical estimator, especially when the number of samples
obtained under one of the two hypotheses is small. The area under the maximum
likelihood estimator is derived; it is a consistent estimator of the true area
under the optimal ROC curve.
