Nonparametric estimation of the threshold at an operating point on the ROC curve

In the problem of binary classification (or medical diagnosis), the classification rule (or diagnostic test) produces a continuous decision variable which is compared to a critical value (or threshold). Test values above (or below) that threshold are called positive (or negative) for disease. The two types of errors associated with every threshold value are Type II (false positive) and Type IIII (false negative) errors. The Receiver Operating Curve (ROC) describes the relationship between probabilities of these two types of errors. The inverse problem is considered; i.e., given the ROC curve (or its estimate) of a particular classification rule, one is interested in finding the value of the threshold ξξ that leads to a specific operating point on that curve. A nonparametric method for estimating the threshold is proposed. Asymptotic distribution is derived for the proposed estimator. Results from simulated data and real-world data are presented for finite sample size. Finding a particular threshold value is crucial in medical diagnoses, among other fields, where a medical test is used to classify a patient as “diseased” or “nondiseased” based on comparing the test result to a particular threshold value. When the ROC is estimated, an operating point is obtained by fixing probability of one type of error, and obtaining the other one from the estimated curve. Threshold estimation can then be viewed as a quantile estimation for one distribution but with the utilization of the second one.