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A receiver operating characteristic (ROC) curve plots the true positive rate of a classier against its false positive rate, both of which are accuracy measures of the classier. The ROC curve has several interesting geometrical properties, including concavity which is a neces- sary condition for a classier to be optimal. In this paper, we study the nonparametric maximum likelihood estimator (NPMLE) of a con- cave ROC curve and its modication to reduce bias. We characterize the NPMLE as a solution to a geometric programming, a special type of a mathematical optimization problem. Wend 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 modication 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 modied NPMLE. Finally, we apply the estimators to estimating the optimal ROC curve of the variance-thresho classier to segment a low depth ofeld image and tonding a diagnostic tool with multiple tests for detection of hemophilia A carrier.

NONPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATION OF A CONCAVE RECEIVER OPERATING CHARACTERISTIC CURVE VIA GEOMETRIC PROGRAMMING