The Generalized Receiver Operating Characteristic Curve

The problem is to predict whether a random outcome is a "success" (R=1) or a "failure" (R=0) given a continuous variable Z. The performance of a prediction rule $D=D(Z)\in \{1,0\}$ boils down to two probabilities, beta =Pr (D=1|R=1) and alpha =Pr (D=1|R=0). We wish beta is high, alpha is low. Given a set of rules D such that any d in D is attributed to a specific alpha, I define the "generalized" receiver operating characteristic (GROC) curve as a function that returns beta for any alpha in (0,1]. The GROC curve associated with D ={d(Z)=I(Z>c),c in R} is the "conventional" ROC curve, while an "efficient" ROC (EROC) curve derives from rules that return the largest possible beta for any alpha in (0,1]. I present estimation theory for the GROC curve and develop procedures for estimating the efficient rules and the associated EROC curve under semiparametric and nonparametric conditions.