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

TL;DR

This paper introduces a new statistical method for ROC curve estimation based on likelihood ratio ordering, using Bernstein polynomials and empirical likelihood, with proven consistency and demonstrated effectiveness through simulations and real data.

Contribution

It proposes a novel Bernstein polynomial approach for ROC estimation under likelihood ratio ordering, with theoretical consistency and improved performance over existing methods.

Findings

The estimators are asymptotically consistent.

The method outperforms competitive approaches in simulations.

Application to real data demonstrates practical utility.

Abstract

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.