Roc Curve Estimation by Sparse Atomic Decomposition

The receiver operating characteristic (ROC) curve is used for classification between two populations. Usually two independent random samples from the populations with absolutely continuous cumulative distribution functions (CDF) F and G are considered. Then the ROC curve is defined as ROC(t) = 1−G(F−1(1− t)) for 0 < t < 1 if the inverse F−1 exists. Classical ROC curve estimator is based on empirical CDF, weighted regression estimator of ROC curve is described in [2]. The estimate based on the best unbiased CDF estimator is considered in [3]. In this contribution we look for an functional approach to the ROC curve estimation. The estimate of ROC(t) − t based on the principle of sparse atomic decomposition will be chosen from the class of linear combinations of such curves under binormal model conditions. In the binormal model we can assume without the loss of generality that F is CDF of N(0, 1) and G is CDF of N(μ, σ). Consequently ROC(t) will be written as ROC(t; μ, σ) to stress the dependency on μ and σ.