Strong approximations for resample quantile processes and application to ROC methodology

The receiver operating characteristic (ROC) curve is defined as true positive rate versus false positive rate obtained by varying a decision threshold criterion. It has been widely used in medical sciences for its ability to measure the accuracy of diagnostic or prognostic tests. Mathematically speaking, ROC curve is the composition of survival function of one population with the quantile function of another population. In this paper, we study strong approximation for the quantile processes of the Bayesian bootstrap (BB) resampling distributions, and use this result to study strong approximations for the BB version of the ROC process in terms of two independent Kiefer processes. The results imply asymptotically accurate coverage probabilities for the confidence bands for the ROC curve and confidence intervals for the area under the curve functional of the ROC constructed using the BB method. Similar results follow for the bootstrap resampling distribution.

Keywords

• Bayesian bootstrap,
• Kiefer process,
• ROC curve,
• strong approximation,
• quantile process