Strong approximations for resample quantile processes and application to ROC methodology

Abstract 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 science 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 bootstrap and the Bayesian bootstrap resampling distributions, and use this result to study strong approximations for the empirical ROC estimator, the corresponding bootstrap, and the Bayesian versions 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 bootstrap and the BB method.