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The receiver operating characteristic (ROC) curve is a powerful statisticaltool and has been widely applied in medical research. In the ROC curveestimation, 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 thebiomarker between the diseased and healthy individuals. With this assumption,we first propose a Bernstein polynomial method to model the distributions ofboth samples; we then estimate the distributions by the maximum empiricallikelihood principle. The ROC curve estimate and the associated summarystatistics are obtained subsequently. Theoretically, we establish theasymptotic consistency of our estimators. Via extensive numerical studies, wecompare the performance of our method with competitive methods. The applicationof our method is illustrated by a real-data example.

Statistical inference for the two-sample problem under likelihood ratio ordering, with application to the ROC curve estimation