A smooth ROC curve estimator based on log-concave density estimates

TL;DR

This paper proposes a new smooth ROC curve estimator based on log-concave density estimates, demonstrating efficiency gains, robustness, and shorter confidence intervals compared to existing methods.

Contribution

The paper introduces a novel ROC curve estimator using log-concave density estimates, showing asymptotic equivalence to empirical curves and improved finite-sample efficiency.

Findings

Efficiency gain over standard empirical ROC in finite samples

Comparable to binormal estimator for normal distributions

Shorter bootstrap confidence intervals with maintained coverage

Abstract

We introduce a new smooth estimator of the ROC curve based on log-concave density estimates of the constituent distributions. We show that our estimate is asymptotically equivalent to the empirical ROC curve if the underlying densities are in fact log-concave. In addition, we empirically show that our proposed estimator exhibits an efficiency gain for finite sample sizes with respect to the standard empirical estimate in various scenarios and that it is only slightly less efficient, if at all, compared to the fully parametric binormal estimate in case the underlying distributions are normal. The estimator is also quite robust against modest deviations from the log-concavity assumption. We show that bootstrap confidence intervals for the value of the ROC curve at a fixed false positive fraction based on the new estimate are on average shorter compared to the approach by Zhou & Qin (2005), while maintaining coverage probability. Computation of our proposed estimate uses the R package logcondens that implements univariate log-concave density estimation and can be done very efficiently using only one line of code. These obtained results lead us to advocate our estimate for a wide range of scenarios.