Time-Dependent ROC Curve Estimation via Bernstein–Schoenberg Smoothing for Right Censored Data

Time-dependent receiver operating characteristic (ROC) analysis is a widely used tool for evaluating prognostic markers in survival settings with right-censored data. While empirical estimators provide a natural baseline, they yield non-smooth step functions. Conventional kernel-based smoothing methods can be sensitive to bandwidth selection and may require transformation-based boundary correction to reduce endpoint bias on the unit interval. The main objective of this study is to develop a smooth nonparametric estimator for the cumulative/dynamic time-dependent ROC curve under right censoring. The proposed method combines an imputation-based empirical ROC framework, which handles unknown event status through imputation weights derived from conditional survival probabilities, with Bernstein–Schoenberg spline operators. The proposed approach provides a flexible smoothing framework, with interior-knot and tail-adaptation parameters selected in a data-driven manner via AIC, that preserves the monotone non-decreasing structure of the ROC curve and remains supported on [0,1], avoiding transformation-based boundary correction. Monte Carlo simulations under log-normal, exponential, and Weibull survival scenarios, varying sample sizes, two censoring rates, and three prediction horizons indicate that the proposed estimator generally achieves lower mean squared error (MSE) than the empirical estimator and shows favorable performance relative to the kernel-based estimator, particularly at early and intermediate prediction horizons. The method is further illustrated through an application to the Veterans’ Administration Lung Cancer Trial dataset, using the Karnofsky score as the prognostic marker, where it showed competitive discrimination performance, yielding AUC estimates comparable to established methods while providing a smooth functional representation.