Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

HOME ALL ARTICLES View

Bull. Korean Math. Soc. 2011; 48(3): 523-537

Printed May 1, 2011

https://doi.org/10.4134/BKMS.2011.48.3.523

Copyright © The Korean Mathematical Society.

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

Kyeong Eun Lee and Johan Lim

Kyungpook National University, Seoul National University

Abstract

A receiver operating characteristic (ROC) curve plots the true positive rate of a classifier against its false positive rate, both of which are accuracy measures of the classifier. The ROC curve has several interesting geometrical properties, including concavity which is a necessary condition for a classifier to be optimal. In this paper, we study the nonparametric maximum likelihood estimator (NPMLE) of a concave ROC curve and its modification to reduce bias. We characterize the NPMLE as a solution to a geometric programming, a special type of a mathematical optimization problem. We find 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 modification of the NPMLE which minimizes the $\mathcal{L}_1$ distance from the empirical ROC curve. We numerically compare the finite sample performance of three estimators, the empirical ROC curve, the NMPLE, and the modified NPMLE. Finally, we apply the estimators to estimating the optimal ROC curve of the variance-threshold classifier to segment a low depth of field image and to finding a diagnostic tool with multiple tests for detection of hemophilia A carrier.

Keywords: concavity, geometric programming, receiver operating characteristic curve

MSC numbers: 62G05