Nonparametric maximum likelihood estimation of a concave receiver operating characteristic curve via geometric programming

A receiver operating characteristic (ROC) curve plots the true positive rate of a classi er against its false positive rate, both of which are accuracy measures of the classi er. The ROC curve has several interesting geometrical properties, including concavity which is a neces- sary condition for a classi er to be optimal. In this paper, we study the nonparametric maximum likelihood estimator (NPMLE) of a con- cave ROC curve and its modi cation to reduce bias. We characterize the NPMLE as a solution to a geometric programming, a special type of a mathematical optimization problem. We nd that the NPMLE is close to the convex hull of the empirical ROC curve and, thus, has smaller variance but positive bias at a given false positive rate. To reduce the bias, we propose a modi cation of the NPMLE which minimizes the L₁ distance from the empirical ROC curve. We numerically compare the nite sample performance of three estimators, the empirical ROC curve, the NMPLE, and the modi ed NPMLE. Finally, we apply the estimators to estimating the optimal ROC curve of the variance-threshold classi er to segment a low depth of eld image and to nding a diagnostic tool with multiple tests for detection of hemophilia A carrier.