The receiver operating characteristic (ROC) curve is a powerful statistical tool and has been widely applied in medical research. In the ROC curve estimation, a commonly used assumption is that larger the biomarker value, greater severity the disease. In this paper, we mathematically interpret ``greater severity of the disease" as ``larger probability of being diseased". This in turn is equivalent to assume the likelihood ratio ordering of the biomarker between the diseased and healthy individuals. With this assumption, we first propose a Bernstein polynomial method to model the distributions of both samples; we then estimate the distributions by the maximum empirical likelihood principle. The ROC curve estimate and the associated summary statistics are obtained subsequently. Theoretically, we establish the asymptotic consistency of our estimators. Via extensive numerical studies, we compare the performance of our method with competitive methods. The application of our method is illustrated by a real-data example.

翻译：接收者操作特征（ROC）曲线是一种强大的统计工具，已广泛应用于医学研究。在ROC曲线估计中，一个常用假设是：生物标志物值越大，疾病严重程度越高。本文从数学角度将“疾病严重程度更高”解释为“患病概率更大”，这等价于假定患病个体与健康个体的生物标志物满足似然比排序。基于该假设，我们首先提出一种Bernstein多项式方法对两组样本的分布进行建模；随后通过最大经验似然原则估计分布函数，并由此获得ROC曲线及其相关汇总统计量的估计。理论上，我们证明了估计量的渐近一致性。通过大量数值研究，我们将所提方法与现有竞争方法进行性能对比，并使用实际数据示例展示了方法的应用效果。