  We introduce a new smooth estimator of the ROC curve based on log-concave
density estimates of the constituent distributions. We show that our estimate
is asymptotically equivalent to the empirical ROC curve if the underlying
densities are in fact log-concave. In addition, we empirically show that our
proposed estimator exhibits an efficiency gain for finite sample sizes with
respect to the standard empirical estimate in various scenarios and that it is
only slightly less efficient, if at all, compared to the fully parametric
binormal estimate in case the underlying distributions are normal. The
estimator is also quite robust against modest deviations from the log-concavity
assumption. We show that bootstrap confidence intervals for the value of the
ROC curve at a fixed false positive fraction based on the new estimate are on
average shorter compared to the approach by Zhou & Qin (2005), while
maintaining coverage probability. Computation of our proposed estimate uses the
R package logcondens that implements univariate log-concave density estimation
and can be done very efficiently using only one line of code. These obtained
results lead us to advocate our estimate for a wide range of scenarios.
